Each guide includes definitions, assumptions, mathematical derivations, references, labeled graphs, and worked examples. Follow the related-subject links to connect the disciplines.
Explore scale → discipline → model. Branches organize all 269 entries; select a model to see its description and relationships. Cross-scale connections appear in each model’s “Relationships to other models” section.
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A broad, expandable atlas of established models and model families. It is not an exhaustive inventory of every model ever published. Scale labels indicate typical use, not strict physical boundaries.
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Showing 269 of 269 entries
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Evolves a nonrelativistic quantum state using a Hamiltonian.
Atomic / molecularPhysical model
Mathematical model & short derivation
Representative formulation
iℏ∂t∂ψ=HψH=−2mℏ2∇2+V
Derivation / construction sketch
Start with the nonrelativistic energy E = p²/(2m) + V.
Represent momentum by −iℏ∇ and energy by iℏ∂/∂t acting on ψ.
Applying these operators gives the time-dependent Schrödinger equation.
Symbols & assumptions
ψ is the wavefunction, m the particle mass, V the potential, and ℏ the reduced Planck constant; the Hamiltonian must include the interactions relevant to the system.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Example 1: Ground-state probability in an infinite well
Ground-state probability in an infinite well
Problem & parameters. A particle is confined by infinite walls at x = 0 and L. Find the normalized ground-state probability density.
L∣ψ1∣2=2sin2(πx/L)
Solution. The walls select ψ = A sin(πx/L). Normalization gives A = √(2/L); square the wavefunction to obtain the plotted density.
Open this section to load its graph, or use the full-size example link below.
Orange point: horizontal coordinate 0.5, calculated vertical coordinate 2. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 2 on the vertical axis. Values are rounded for display.
Scope. Analytical solution or constitutive evaluation for the stated special case. It is not a general solution of the full model family.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Require anticommuting matrices α and β so the cross terms cancel.
This matrix structure makes ψ a spinor and yields the Dirac Hamiltonian.
Symbols & assumptions
Free-particle form; α and β are Dirac matrices, c the speed of light. Electromagnetic coupling requires potentials and p → p − qA.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. For a free massive Dirac particle, evaluate the positive-energy branch versus momentum.
E/(mc2)=1+(p/mc)2
Solution. Squaring the free Dirac Hamiltonian gives E² = m²c⁴+p²c². Select its positive root.
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Orange point: horizontal coordinate 1.5, calculated vertical coordinate 1.8028. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 1.5, into the displayed formula to obtain 1.8028 on the vertical axis. Values are rounded for display.
Scope. Analytical solution or constitutive evaluation for the stated special case. It is not a general solution of the full model family.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Separates electronic motion from slower nuclear motion.
Atomic / molecularPhysical model
Mathematical model & short derivation
Representative formulation
Ψ(r,R)≈ψn(r;R)χn(R)He(R)ψn=En(R)ψn
Derivation / construction sketch
Write the total Hamiltonian as nuclear kinetic energy plus an electronic Hamiltonian at fixed nuclear coordinates R.
Expand the full state in electronic eigenstates.
Neglect couplings generated by nuclear derivatives of the electronic states to obtain motion on one potential-energy surface Eₙ(R).
Symbols & assumptions
r and R denote electronic and nuclear coordinates; the approximation can fail near electronic degeneracies or rapid nonadiabatic transitions.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Example 1: Nuclear motion on a harmonic energy surface
Nuclear motion on a harmonic energy surface
Problem & parameters. Approximate one Born–Oppenheimer potential-energy surface near its minimum by a spring of stiffness k.
(U−U0)/(kℓ2)=21q2,q=(R−Re)/ℓ
Solution. Taylor-expand the electronic energy about its minimum. The linear term vanishes; retain the quadratic term.
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Orange point: horizontal coordinate 0, calculated vertical coordinate 0. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 0 on the vertical axis. Values are rounded for display.
Scope. Local harmonic approximation on a single adiabatic surface; electronic crossings and nonadiabatic coupling are excluded.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Approximates a many-electron wavefunction by one self-consistent Slater determinant.
Atomic / molecularPhysical model
Mathematical model & short derivation
Representative formulation
Fϕi=εiϕiF=h+j∑(Jj−Kj)
Derivation / construction sketch
Approximate the many-electron state by one antisymmetrized determinant.
Minimize its energy subject to orbital orthonormality using Lagrange multipliers.
Variation with respect to each orbital produces the Fock equation, solved self-consistently.
Symbols & assumptions
Spin-orbital form; h is the one-electron operator, J Coulomb and K exchange. Electron correlation beyond exchange is omitted.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Practical applications & products
Application area
Quantum & electronic structure
Practical use
Estimating molecular orbitals before a correlation calculation.
Example 1: One-electron hydrogenic radial probability
One-electron hydrogenic radial probability
Problem & parameters. Use the normalized hydrogen 1s state for one electron in a Coulomb potential. Plot probability per radial interval.
a0P(r)=4(r/a0)2e−2r/a0
Solution. The 1s density is exp(−2r/a₀)/(πa₀³). Multiply by the spherical volume factor 4πr².
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Orange point: horizontal coordinate 3, calculated vertical coordinate 0.089235. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 3, into the displayed formula to obtain 0.089235 on the vertical axis. Values are rounded for display.
Scope. Hartree–Fock is exact for this one-electron case. For DFT this is an exact-functional reference; approximate functionals need not reproduce it exactly.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Express the ground-state energy as a density functional including kinetic, external, Hartree and exchange-correlation terms.
Represent the noninteracting kinetic term using orbitals, with n(r) = Σᵢ fᵢ∣φᵢ(r)∣².
Vary the orbitals under orthonormality constraints to obtain the Kohn–Sham equations.
Symbols & assumptions
fᵢ are orbital occupations. vxc = δExc/δn is approximated in practical calculations.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Example 1: One-electron hydrogenic radial probability
One-electron hydrogenic radial probability
Problem & parameters. Use the normalized hydrogen 1s state for one electron in a Coulomb potential. Plot probability per radial interval.
a0P(r)=4(r/a0)2e−2r/a0
Solution. The 1s density is exp(−2r/a₀)/(πa₀³). Multiply by the spherical volume factor 4πr².
Open this section to load its graph, or use the full-size example link below.
Orange point: horizontal coordinate 3, calculated vertical coordinate 0.089235. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 3, into the displayed formula to obtain 0.089235 on the vertical axis. Values are rounded for display.
Scope. Hartree–Fock is exact for this one-electron case. For DFT this is an exact-functional reference; approximate functionals need not reproduce it exactly.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Evolves electron density to approximate excited-state response.
Atomic / molecularPhysical model
Mathematical model & short derivation
Representative formulation
iℏ∂t∂ϕi=[−2meℏ2∇2+vs[n](r,t)]ϕi
Derivation / construction sketch
Map the interacting time-dependent density onto an auxiliary noninteracting orbital system.
Require the effective potential vs to reproduce that density.
Propagating the orbitals yields density and response; linearizing around a stationary state gives excitation-response equations.
Symbols & assumptions
Time-dependent Kohn–Sham form; practical exchange-correlation potentials often neglect memory and may misrepresent charge-transfer excitations.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Consider a resonantly driven, noninteracting two-level reference starting in its lower state.
P2(t)=sin2(Ωt/2)
Solution. Solve the resonant two-amplitude system to obtain upper-state amplitude −i sin(Ωt/2), then take its squared magnitude.
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Orange point: horizontal coordinate 3.1416, calculated vertical coordinate 1. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 3.1416, into the displayed formula to obtain 1 on the vertical axis. Values are rounded for display.
Scope. Two-level rotating-wave reference for time-dependent electronic calculations; not a general TDDFT solution.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Represents electronic states with localized orbitals and hopping parameters.
Atomic / molecularPhysical model
Mathematical model & short derivation
Representative formulation
H=i∑εici†ci+ij∑tijci†cj
Derivation / construction sketch
Expand electronic states in localized atomic-like orbitals.
Project the electronic Hamiltonian into that basis.
Retain selected on-site and hopping matrix elements to obtain a finite matrix eigenproblem.
Symbols & assumptions
cᵢ† and cᵢ create and remove electrons at orbital i; εᵢ and tᵢⱼ are fitted or computed energies. Overlap requires a generalized eigenproblem if the basis is nonorthogonal.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. An infinite one-orbital chain has nearest-neighbor hopping tₕ and zero on-site energy.
E(k)/th=−2cos(ka)
Solution. Insert a Bloch state exp(ikna) into the hopping equation; the two neighbors contribute −tₕ(exp(ika)+exp(−ika)).
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Orange point: horizontal coordinate 0, calculated vertical coordinate -2. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain -2 on the vertical axis. Values are rounded for display.
Scope. Analytical solution or constitutive evaluation for the stated special case. It is not a general solution of the full model family.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Models competition between particle hopping and local electron interactions.
Atomic / molecularPhysical model
Mathematical model & short derivation
Representative formulation
H=−t⟨i,j⟩,σ∑(ciσ†cjσ+h.c.)+Ui∑ni↑ni↓
Derivation / construction sketch
Start from a localized-orbital description with electron-electron repulsion.
Retain nearest-neighbor hopping and only the dominant on-site repulsion.
Opposite-spin occupancy of one site then costs energy U.
Symbols & assumptions
Single-band Hubbard form; t is hopping energy, U on-site repulsion, n occupation, and h.c. the Hermitian conjugate.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Find the two-electron singlet ground energy of a two-site Hubbard dimer with hopping tₕ > 0 and repulsion U.
E0/th=21[u−u2+16],u=U/th
Solution. In the coupled singlet/double-occupancy block the matrix has diagonal 0,U and off-diagonal −2tₕ. Solve its quadratic characteristic equation and select the lower eigenvalue.
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Orange point: horizontal coordinate 6, calculated vertical coordinate -0.60555. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 6, into the displayed formula to obtain -0.60555 on the vertical axis. Values are rounded for display.
Scope. Analytical solution or constitutive evaluation for the stated special case. It is not a general solution of the full model family.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Represents interacting localized magnetic moments.
Atomic / molecularPhysical model
Mathematical model & short derivation
Representative formulation
H=−⟨i,j⟩∑JijSi⋅Sj−gμBB⋅i∑Si
Derivation / construction sketch
Restrict the low-energy degrees of freedom to localized magnetic moments.
Represent rotationally invariant pair coupling by a scalar product.
Add the Zeeman interaction with an external field.
Symbols & assumptions
Sᵢ are dimensionless spin operators; positive J favors parallel spins under this sign convention. g is the Landé factor and μB the Bohr magneton.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Two classical unit spins interact through −J s₁·s₂ with J > 0.
E/J=−cosθ
Solution. The dot product of two unit vectors is cos θ. Parallel alignment minimizes the energy.
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Orange point: horizontal coordinate 1.5708, calculated vertical coordinate -6.1232e-17. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 1.5708, into the displayed formula to obtain -6.1232e-17 on the vertical axis. Values are rounded for display.
Scope. Classical two-spin special case; quantum spin spectra require a different treatment.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Represents discrete spins with interaction energies.
Atomic / molecularPhysical model
Mathematical model & short derivation
Representative formulation
H=−J⟨i,j⟩∑sisj−hi∑sisi=±1
Derivation / construction sketch
Restrict each local moment to two orientations along one axis.
Assign an interaction energy to neighboring pairs and a field energy to each spin.
Summing these contributions gives the Ising Hamiltonian.
Symbols & assumptions
J and h have energy units; dimensionality, interaction range and boundary conditions change the predicted behavior.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Practical applications & products
Application area
Quantum & electronic structure
Practical use
Studying phase transitions in a simplified magnet.
Problem & parameters. A single spin s = ±1 has energy −hs at inverse temperature β. Find its thermal mean.
⟨s⟩=tanh(βh)
Solution. Its partition function is 2 cosh(βh). The weighted spin sum is 2 sinh(βh); divide to obtain tanh(βh).
Open this section to load its graph, or use the full-size example link below.
Orange point: horizontal coordinate 0, calculated vertical coordinate 0. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 0 on the vertical axis. Values are rounded for display.
Scope. Analytical solution or constitutive evaluation for the stated special case. It is not a general solution of the full model family.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Describes a quantum degree of freedom in a quadratic potential.
Atomic / molecularPhysical model
Mathematical model & short derivation
Representative formulation
H=2mp2+21mω2x2En=ℏω(n+21)
Derivation / construction sketch
Expand a smooth potential about a stable minimum to quadratic order.
Rewrite the quantum Hamiltonian using raising and lowering operators.
Their commutator gives equally spaced eigenvalues and a nonzero ground-state energy.
Symbols & assumptions
n = 0,1,…; x measures displacement from equilibrium and ω is the natural frequency. Anharmonic terms are neglected.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Practical applications & products
Application area
Quantum & electronic structure
Practical use
Approximating vibrational levels near equilibrium.
Problem & parameters. Use oscillator length ℓ = √(ℏ/mω) and find the normalized ground-state density.
ℓ∣ψ0∣2=π−1/2e−(x/ℓ)2
Solution. Substitute a Gaussian into the stationary Schrödinger equation. The ground-state wavefunction is exp(−x²/2ℓ²)/(π¼√ℓ); square it.
Open this section to load its graph, or use the full-size example link below.
Orange point: horizontal coordinate 0, calculated vertical coordinate 0.56419. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 0.56419 on the vertical axis. Values are rounded for display.
Scope. Analytical solution or constitutive evaluation for the stated special case. It is not a general solution of the full model family.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Confines a quantum particle within idealized boundaries.
Atomic / molecularPhysical model
Mathematical model & short derivation
Representative formulation
ψn(x)=L2sinLnπxEn=2mL2n2π2ℏ2
Derivation / construction sketch
Solve the stationary free-particle Schrödinger equation inside a one-dimensional box.
Impose ψ(0) = ψ(L) = 0, which selects k = nπ/L.
Normalize the sine functions and substitute k into E = ℏ²k²/(2m).
Symbols & assumptions
Infinite-wall box, 0 < x < L, n = 1,2,…; finite barriers give different states and allow penetration outside the box.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Practical applications & products
Application area
Quantum & electronic structure
Practical use
Understanding size-dependent electronic energy levels.
Example 1: Ground-state probability in an infinite well
Ground-state probability in an infinite well
Problem & parameters. A particle is confined by infinite walls at x = 0 and L. Find the normalized ground-state probability density.
L∣ψ1∣2=2sin2(πx/L)
Solution. The walls select ψ = A sin(πx/L). Normalization gives A = √(2/L); square the wavefunction to obtain the plotted density.
Open this section to load its graph, or use the full-size example link below.
Orange point: horizontal coordinate 0.5, calculated vertical coordinate 2. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 2 on the vertical axis. Values are rounded for display.
Scope. Analytical solution or constitutive evaluation for the stated special case. It is not a general solution of the full model family.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Integrates atomic motion under specified interaction forces.
Atomic / molecularPhysical model
Mathematical model & short derivation
Representative formulation
midt2d2ri=−∇iU(r1,…,rN)
Derivation / construction sketch
Specify a potential energy U for the atomic configuration.
Differentiate U with respect to each position to obtain force.
Apply Newton’s second law and integrate positions and velocities in time.
Symbols & assumptions
rᵢ and mᵢ are atomic positions and masses. Thermostats, constraints and long-range electrostatics modify the practical equations or integration.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Take one isolated coordinate with potential kq²/2, initial displacement A, and zero initial velocity.
q(τ)=cosτ
Solution. Newton’s equation reduces to q″+ω²q = 0. The initial data select A cos(ωt).
Open this section to load its graph, or use the full-size example link below.
Orange point: horizontal coordinate 3.1416, calculated vertical coordinate -1. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 3.1416, into the displayed formula to obtain -1 on the vertical axis. Values are rounded for display.
Scope. Harmonic force benchmark for MD or locally harmonic ab initio dynamics; real many-atom trajectories are not generally sinusoidal.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Computes interatomic forces from electronic-structure calculations during motion.
Atomic / molecularPhysical model
Mathematical model & short derivation
Representative formulation
MIR¨I=−∇IEelec(R)
Derivation / construction sketch
At each nuclear configuration, solve an electronic-structure problem.
Use its converged energy as the nuclear potential-energy surface.
Differentiate that energy to obtain nuclear forces and integrate the nuclei classically.
Symbols & assumptions
Born–Oppenheimer MD form; Mᴵ and Rᴵ are nuclear masses and positions. Electronic convergence and force consistency matter.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Take one isolated coordinate with potential kq²/2, initial displacement A, and zero initial velocity.
q(τ)=cosτ
Solution. Newton’s equation reduces to q″+ω²q = 0. The initial data select A cos(ωt).
Open this section to load its graph, or use the full-size example link below.
Orange point: horizontal coordinate 3.1416, calculated vertical coordinate -1. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 3.1416, into the displayed formula to obtain -1 on the vertical axis. Values are rounded for display.
Scope. Harmonic force benchmark for MD or locally harmonic ab initio dynamics; real many-atom trajectories are not generally sinusoidal.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Combines short-range repulsion with an inverse-sixth-power attraction.
Atomic / molecularPhysical model
Mathematical model & short derivation
Representative formulation
U(r)=4ε[(σ/r)12−(σ/r)6]F(r)=−drdU
Derivation / construction sketch
Model dispersion attraction by −C₆/r⁶.
Approximate short-range repulsion by a steeper r⁻¹² term.
Choose ε and σ to set the well depth and zero crossing, then differentiate for force.
Symbols & assumptions
r is pair separation, ε the energy scale and σ the length scale; this pair model is not a general description of chemical bonding.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Evaluate an unshifted 12–6 pair potential at reduced separation r/σ.
U/ε=4[(σ/r)12−(σ/r)6]
Solution. Insert the reduced distance into the two inverse powers. Differentiating gives a minimum at r/σ = 2^(1/6), with U/ε = −1.
Open this section to load its graph, or use the full-size example link below.
Orange point: horizontal coordinate 1.975, calculated vertical coordinate -0.066264. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 1.975, into the displayed formula to obtain -0.066264 on the vertical axis. Values are rounded for display.
Scope. For water and Martini entries, this is only a Lennard–Jones interaction contribution; electrostatics, constraints, and other sites are not included.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Represents an anharmonic bond with a finite dissociation energy.
Atomic / molecularPhysical model
Mathematical model & short derivation
Representative formulation
U(r)=De[1−exp(−a(r−re))]2
Derivation / construction sketch
Seek a bond potential with a minimum at rₑ and finite dissociation energy.
Square an exponential displacement expression to obtain both properties.
A small-displacement expansion gives U ≈ Dₑa²(r − rₑ)² and spring constant k = 2Dₑa².
Symbols & assumptions
Dₑ is well depth, a inverse length; this convention sets U(rₑ) = 0 and U(∞) = Dₑ.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Evaluate a Morse bond with its dissociation limit set to zero.
U/De=[1−e−q]2−1,q=a(r−re)
Solution. At q = 0 the energy is −Dₑ. As q increases, the exponential tends to zero and the energy approaches the dissociation limit.
Open this section to load its graph, or use the full-size example link below.
Orange point: horizontal coordinate 1.7, calculated vertical coordinate -0.33199. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 1.7, into the displayed formula to obtain -0.33199 on the vertical axis. Values are rounded for display.
Scope. Analytical solution or constitutive evaluation for the stated special case. It is not a general solution of the full model family.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Approximate the local electronic environment by a sum of neighbor density contributions.
Assign an embedding cost F to inserting an atom into that environment.
Add pair interactions and differentiate the total energy to obtain many-body forces.
Symbols & assumptions
F is embedding energy, φ pair energy and f a density contribution; parameters are element- and alloy-specific.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Example 1: Illustrative embedding-energy contribution
Illustrative embedding-energy contribution
Problem & parameters. Choose the illustrative embedding function F = −E*√(ρ/ρ*). Plot its density dependence.
F(ρ)/E∗=−ρ/ρ∗
Solution. Substitute the normalized local density into the chosen function. This evaluates the embedding contribution before summing pair terms.
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Orange point: horizontal coordinate 2.005, calculated vertical coordinate -1.416. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 2.005, into the displayed formula to obtain -1.416 on the vertical axis. Values are rounded for display.
Scope. Illustrative EAM-type embedding function; not a fitted material parameterization. MEAM angular screening and density corrections are held fixed.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Extends embedding models with angular information.
Atomic / molecularPhysical model
Mathematical model & short derivation
Representative formulation
U=i∑Fi(ρˉi)+21i=j∑Sijϕij(rij)
Derivation / construction sketch
Begin with the embedded-atom energy.
Construct an effective density ρ̄ that includes angular information.
Use screening S to account for the local environment of a bond.
Symbols & assumptions
Representative MEAM structure; angular-density definitions and screening functions depend on the parameterization.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Example 1: Illustrative embedding-energy contribution
Illustrative embedding-energy contribution
Problem & parameters. Choose the illustrative embedding function F = −E*√(ρ/ρ*). Plot its density dependence.
F(ρ)/E∗=−ρ/ρ∗
Solution. Substitute the normalized local density into the chosen function. This evaluates the embedding contribution before summing pair terms.
Open this section to load its graph, or use the full-size example link below.
Orange point: horizontal coordinate 2.005, calculated vertical coordinate -1.416. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 2.005, into the displayed formula to obtain -1.416 on the vertical axis. Values are rounded for display.
Scope. Illustrative EAM-type embedding function; not a fitted material parameterization. MEAM angular screening and density corrections are held fixed.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Makes bond strength depend on the local bonding environment.
Atomic / molecularPhysical model
Mathematical model & short derivation
Representative formulation
U=21i=j∑fc(rij)[fR(rij)+bijfA(rij)]
Derivation / construction sketch
Split pair contributions into repulsive and attractive parts.
Let the attractive bond strength b depend on coordination and bond angles.
Apply a cutoff fc and sum to obtain environment-dependent bonding.
Symbols & assumptions
fR and fA are radial terms, b is bond order. A full parameter set is required for a particular material.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. In a Tersoff-form pair term, hold cutoff and bond order at one and choose two exponential terms with coefficients 1 and 2.
U/E∗=e−2q−2e−q
Solution. Substitute the fixed bond order into the repulsive-minus-attractive energy. Differentiate the resulting two exponentials to inspect the force.
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Orange point: horizontal coordinate 2, calculated vertical coordinate -0.25235. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 2, into the displayed formula to obtain -0.25235 on the vertical axis. Values are rounded for display.
Scope. Toy fixed-environment pair contribution; this excludes environment-dependent bond order and cutoff transitions.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Uses two-body and three-body terms to favor local tetrahedral structure.
Atomic / molecularPhysical model
Mathematical model & short derivation
Representative formulation
U=i<j∑V2(rij)+i,j<k∑V3(rij,rik,θjik)
Derivation / construction sketch
Start with radial pair interactions.
Add an angular energy penalizing departures from a preferred bond geometry.
Sum pair and three-body contributions to obtain forces that favor the chosen local structure.
Symbols & assumptions
Representative Stillinger–Weber decomposition; V₃ often contains an angular square and radial cutoff factors.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Hold the radial factor of a Stillinger–Weber three-body term fixed and vary the included angle.
U3/K=(cosθ+1/3)2
Solution. The squared angular factor vanishes at cos θ = −1/3, giving the tetrahedral angle.
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Orange point: horizontal coordinate 1.5708, calculated vertical coordinate 0.11111. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 1.5708, into the displayed formula to obtain 0.11111 on the vertical axis. Values are rounded for display.
Scope. Angular contribution only, with fixed radial prefactor K > 0.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Uses variable bond orders and charge equilibration to represent chemical reactions.
Atomic / molecularPhysical model
Mathematical model & short derivation
Representative formulation
U=Ubond(BO)+Uangle+Utorsion+UvdW+UCoulomb+⋯
Derivation / construction sketch
Represent bond order BO as a continuous function of atom separations.
Use bond orders to adjust bonded energies as coordination changes.
Combine them with nonbonded and charge-equilibration contributions to obtain a reactive potential.
Symbols & assumptions
Schematic ReaxFF energy decomposition, not a complete implementation. Chemistry-specific terms and parameters are essential.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Near a stable isolated bond minimum, use the local quadratic energy with curvature k > 0.
ΔU/(kℓ2)=q2/2
Solution. The energy gradient vanishes at equilibrium. Retaining the second Taylor derivative gives ΔU = k(Δr)²/2.
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Orange point: horizontal coordinate 0, calculated vertical coordinate 0. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 0 on the vertical axis. Values are rounded for display.
Scope. Local Taylor benchmark, not the full force field or a trained potential prediction; reactive changes and other coordinates are held fixed.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Represent bond stretching and angle bending by harmonic expansions.
Use periodic Fourier terms for torsions.
Add electrostatic and van der Waals interactions for nonbonded pairs.
Symbols & assumptions
Representative AMBER form; numerical prefactors, 1–4 scaling and parameters must follow the selected version. Unb denotes nonbonded energy.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Near a stable isolated bond minimum, use the local quadratic energy with curvature k > 0.
ΔU/(kℓ2)=q2/2
Solution. The energy gradient vanishes at equilibrium. Retaining the second Taylor derivative gives ΔU = k(Δr)²/2.
Open this section to load its graph, or use the full-size example link below.
Orange point: horizontal coordinate 0, calculated vertical coordinate 0. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 0 on the vertical axis. Values are rounded for display.
Scope. Local Taylor benchmark, not the full force field or a trained potential prediction; reactive changes and other coordinates are held fixed.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Models biomolecular interactions with chemistry-specific parameter sets.
Atomic / molecularPhysical model
Mathematical model & short derivation
Representative formulation
U=Ubond+Uangle+Udihedral+Uimproper+UUB+Unb
Derivation / construction sketch
Expand local molecular distortions around fitted geometries.
Add periodic torsions and improper terms to maintain stereochemistry.
Include Urey–Bradley distance terms and nonbonded interactions where specified.
Symbols & assumptions
Representative CHARMM family form; selected force fields may add CMAP or polarization terms. UUB is a 1–3 distance contribution.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Near a stable isolated bond minimum, use the local quadratic energy with curvature k > 0.
ΔU/(kℓ2)=q2/2
Solution. The energy gradient vanishes at equilibrium. Retaining the second Taylor derivative gives ΔU = k(Δr)²/2.
Open this section to load its graph, or use the full-size example link below.
Orange point: horizontal coordinate 0, calculated vertical coordinate 0. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 0 on the vertical axis. Values are rounded for display.
Scope. Local Taylor benchmark, not the full force field or a trained potential prediction; reactive changes and other coordinates are held fixed.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Describe rotation around a bond by a periodic energy function.
Expand it in a cosine series with OPLS phase conventions.
Fit the coefficients and combine the torsion with bonded and nonbonded energy terms.
Symbols & assumptions
Representative OPLS torsional term; it is one component of the force field, with version-specific charges, combining rules and pair scaling.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Retain only the first OPLS torsion coefficient V₁.
U/V1=21(1+cosϕ)
Solution. Set the other Fourier coefficients to zero and evaluate the remaining cosine term.
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Orange point: horizontal coordinate 3.1416, calculated vertical coordinate 0. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 3.1416, into the displayed formula to obtain 0 on the vertical axis. Values are rounded for display.
Scope. Single torsional energy contribution, not the full molecular force field.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Place charges on oxygen and hydrogens and a Lennard–Jones site on oxygen.
Sum intermolecular electrostatic and dispersion-repulsion terms; SPC/E also includes a mean polarization-energy correction.
Symbols & assumptions
Only intermolecular terms are shown; rigid geometry, charges and the SPC/E correction must use the published parameter set.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Evaluate an unshifted 12–6 pair potential at reduced separation r/σ.
U/ε=4[(σ/r)12−(σ/r)6]
Solution. Insert the reduced distance into the two inverse powers. Differentiating gives a minimum at r/σ = 2^(1/6), with U/ε = −1.
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Orange point: horizontal coordinate 1.975, calculated vertical coordinate -0.066264. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 1.975, into the displayed formula to obtain -0.066264 on the vertical axis. Values are rounded for display.
Scope. For water and Martini entries, this is only a Lennard–Jones interaction contribution; electrostatics, constraints, and other sites are not included.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Uses a four-site geometry with an off-oxygen charge site.
Atomic / molecularPhysical model
Mathematical model & short derivation
Representative formulation
U=i<j∑4πε0rijqiqj+UOO,LJqO=0
Derivation / construction sketch
Separate the Lennard–Jones oxygen site from the negative-charge site M.
Place positive charges on the two hydrogens and negative charge on M.
Evaluate intermolecular Coulomb and oxygen-oxygen Lennard–Jones energies.
Symbols & assumptions
Rigid four-site TIP4P family; M position, charges and parameters vary across TIP4P variants.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Practical applications & products
Application area
Molecular dynamics & force fields
Practical use
Liquid-water and ice studies with a chosen parameter variant.
Problem & parameters. Evaluate an unshifted 12–6 pair potential at reduced separation r/σ.
U/ε=4[(σ/r)12−(σ/r)6]
Solution. Insert the reduced distance into the two inverse powers. Differentiating gives a minimum at r/σ = 2^(1/6), with U/ε = −1.
Open this section to load its graph, or use the full-size example link below.
Orange point: horizontal coordinate 1.975, calculated vertical coordinate -0.066264. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 1.975, into the displayed formula to obtain -0.066264 on the vertical axis. Values are rounded for display.
Scope. For water and Martini entries, this is only a Lennard–Jones interaction contribution; electrostatics, constraints, and other sites are not included.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Uses auxiliary charged particles to represent induced polarization.
Atomic / molecularPhysical model
Mathematical model & short derivation
Representative formulation
UDrude=21kDd2−qDd⋅Eα=kDqD2
Derivation / construction sketch
Attach an auxiliary charge qD to an atom by a harmonic spring.
Minimize its energy in an electric field, giving kD d = qD E.
The induced dipole p = qD d is therefore αE.
Symbols & assumptions
d is displacement, kD spring stiffness and α polarizability in consistent units; short-range damping and thermostat treatment are often required.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. A charged Drude oscillator has harmonic stiffness k and charge q. Find its static induced dipole.
p/(αE∗)=E/E∗
Solution. Balance kx = qE. Then p = qx = (q²/k)E, so α = q²/k.
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Orange point: horizontal coordinate 0, calculated vertical coordinate 0. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 0 on the vertical axis. Values are rounded for display.
Scope. Analytical solution or constitutive evaluation for the stated special case. It is not a general solution of the full model family.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
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Fits atomic energies and forces from reference data using statistical learning.
Atomic / molecularPhysical model
Mathematical model & short derivation
Representative formulation
Eθ(R)=i∑εθ(Di(R))Fi=−∂ri∂Eθ
Derivation / construction sketch
Encode each atomic neighborhood with descriptors Dᵢ or learned equivariant features.
Fit energy and force predictions to reference calculations.
Differentiate the fitted energy to enforce conservative forces.
Symbols & assumptions
Representative local machine-learning potential; θ are trained parameters. Long-range effects and out-of-distribution configurations need additional treatment.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Practical applications & products
Application area
Molecular dynamics & force fields
Practical use
Accelerating repeated atomistic calculations within a validated training domain.
Problem & parameters. Near a stable isolated bond minimum, use the local quadratic energy with curvature k > 0.
ΔU/(kℓ2)=q2/2
Solution. The energy gradient vanishes at equilibrium. Retaining the second Taylor derivative gives ΔU = k(Δr)²/2.
Open this section to load its graph, or use the full-size example link below.
Orange point: horizontal coordinate 0, calculated vertical coordinate 0. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 0 on the vertical axis. Values are rounded for display.
Scope. Local Taylor benchmark, not the full force field or a trained potential prediction; reactive changes and other coordinates are held fixed.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Map atomistic coordinates r to coarse coordinates R through M.
Integrate the microscopic Boltzmann distribution over the eliminated coordinates.
Take its negative logarithm to define a potential of mean force.
Symbols & assumptions
Exact equilibrium coarse graining is generally many-body and state-dependent; practical pair approximations lose some information.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Let a coarse variable have Gaussian probability proportional to exp(−q²/2).
F(q)/(kBT)=q2/2
Solution. Apply F = −kBT ln P, and remove the additive normalization constant.
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Orange point: horizontal coordinate 0, calculated vertical coordinate 0. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 0 on the vertical axis. Values are rounded for display.
Scope. Exactly solvable Gaussian coarse-graining example; it does not assert that arbitrary coarse models are harmonic.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Uses mapped molecular beads and parameterized interactions.
Micro / mesoPhysical model
Mathematical model & short derivation
Representative formulation
U=Ubonded+i<j∑[ULJ(rij)+UCoulomb(rij)]
Derivation / construction sketch
Map groups of atoms to bead types.
Assign bead interactions to reproduce selected thermodynamic and structural targets.
Combine nonbonded bead interactions with mapped bonded terms.
Symbols & assumptions
Representative Martini architecture; mapping, bead types, interaction matrices and electrostatics depend on the selected version.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Evaluate an unshifted 12–6 pair potential at reduced separation r/σ.
U/ε=4[(σ/r)12−(σ/r)6]
Solution. Insert the reduced distance into the two inverse powers. Differentiating gives a minimum at r/σ = 2^(1/6), with U/ε = −1.
Open this section to load its graph, or use the full-size example link below.
Orange point: horizontal coordinate 1.975, calculated vertical coordinate -0.066264. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 1.975, into the displayed formula to obtain -0.066264 on the vertical axis. Values are rounded for display.
Scope. For water and Martini entries, this is only a Lennard–Jones interaction contribution; electrostatics, constraints, and other sites are not included.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Example 1: Mean relative velocity under fixed pair drag
Mean relative velocity under fixed pair drag
Problem & parameters. Hold pair distance and weight fixed; the mean relative velocity obeys dy/dτ = −y. Random force has zero mean.
y(τ)=e−τ
Solution. Separate dy/dτ = −y, integrate ln(y) = −τ + C, and apply y(0) = 1.
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Orange point: horizontal coordinate 2.5, calculated vertical coordinate 0.082085. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.082085 on the vertical axis. Values are rounded for display.
Scope. Mean of a linear frozen-geometry pair reduction. DPD sample trajectories fluctuate and require a stochastic integrator.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
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Uses overdamped stochastic motion for particles in a surrounding medium.
Micro / mesoPhysical model
Mathematical model & short derivation
Representative formulation
dri=μiFidt+2DidWiDi=μikBT
Derivation / construction sketch
Start from Langevin motion with rapid momentum relaxation.
Neglect inertia on time scales long compared with m/ζ.
Balance drift and thermal diffusion to obtain overdamped motion.
Symbols & assumptions
Independent constant mobility μᵢ shown; hydrodynamic interactions or position-dependent mobility introduce matrix diffusion and additional drift.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Example 1: One-dimensional mean-square displacement
One-dimensional mean-square displacement
Problem & parameters. For free Brownian motion in one dimension take D = 1 m²/s and initial position zero.
⟨[x(t)−x(0)]2⟩=2Dt
Solution. Integrate dx = √(2D)dW. Since the variance of W(t) is t, the mean-square displacement is 2Dt.
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Orange point: horizontal coordinate 2.5, calculated vertical coordinate 5. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 5 on the vertical axis. Values are rounded for display.
Scope. Ensemble expectation, not a single random trajectory; illustrative diffusivity.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Adds friction and random forces to a dynamical model.
Micro / mesoPhysical model
Mathematical model & short derivation
Representative formulation
mv˙=F−ζv+η(t)⟨ηa(t)ηb(t′)⟩=2ζkBTδabδ(t−t′)
Derivation / construction sketch
Separate resolved forces from fast environmental effects.
Approximate the latter as linear friction and white noise.
Choose the noise covariance so the equilibrium velocity distribution has temperature T.
Symbols & assumptions
ζ is friction; noise is idealized as memoryless and Gaussian. Generalized Langevin models retain memory kernels.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. A free Langevin particle has linear drag γ, mass m, mean initial speed v₀, and zero-mean thermal noise. Use τ = γt/m.
y(τ)=e−τ
Solution. Separate dy/dτ = −y, integrate ln(y) = −τ + C, and apply y(0) = 1.
Open this section to load its graph, or use the full-size example link below.
Orange point: horizontal coordinate 2.5, calculated vertical coordinate 0.082085. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.082085 on the vertical axis. Values are rounded for display.
Scope. Ensemble mean velocity; the plotted smooth decay is not an individual noisy trajectory.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
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Samples transitions between states using event rates.
Micro / mesoNumerical method
Mathematical model & short derivation
Representative formulation
Δt=−Klnu1P(event i)=KkiK=i∑ki
Derivation / construction sketch
Assume independent exponential waiting times for allowed events.
The probability that no event occurs before t is exp(−Kt).
Invert that survival distribution and select an event in proportion to its rate.
Symbols & assumptions
u₁ is uniform on (0,1); rates kᵢ must represent the relevant Markov transitions.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. A kinetic Monte Carlo process has one constant total escape rate λ. Find the probability that its first event has occurred.
P(T≤t)=1−e−λt
Solution. The survival probability solves S′ = −λS with S(0) = 1. Subtract S from one.
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Orange point: horizontal coordinate 2.5, calculated vertical coordinate 0.91792. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.91792 on the vertical axis. Values are rounded for display.
Scope. Waiting-time distribution for a fixed state and rate, not the entire evolving event network.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
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Evolves a conserved composition field through chemical-potential gradients.
Micro / mesoPhysical model
Mathematical model & short derivation
Representative formulation
∂t∂c=∇⋅(M∇μ)μ=f′(c)−κ∇2c
Derivation / construction sketch
Define free energy F = ∫[f(c)+κ∣∇c∣²/2]dV.
Take its variational derivative to obtain chemical potential μ.
Use flux J = −M∇μ in composition conservation ∂tc = −∇·J.
Symbols & assumptions
c is conserved composition, M mobility and κ gradient-energy coefficient; boundary conditions determine mass and energy behavior.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Use dimensionless Cahn–Hilliard dynamics with M = a = κ = 1, quadratic free energy ac²/2, periodic boundaries, and initial perturbation cos x. Plot t = 1.
c−cˉ=e−2cosx
Solution. For wave number one, the amplitude satisfies A′ = −M(a+κ)A = −2A. Thus A(1) = exp(−2).
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Orange point: horizontal coordinate 3.1416, calculated vertical coordinate -0.13534. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 3.1416, into the displayed formula to obtain -0.13534 on the vertical axis. Values are rounded for display.
Scope. Exact quadratic-free-energy special case, not nonlinear phase separation.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
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Evolves a nonconserved order parameter toward lower free energy.
Micro / mesoPhysical model
Mathematical model & short derivation
Representative formulation
∂t∂η=−LδηδFF=∫[f(η)+κ∣∇η∣2/2]dV
Derivation / construction sketch
Use an order parameter η that need not be conserved.
Choose local gradient descent of the free energy.
This gives ∂tη = −L[f′(η)−κ∇²η], which reduces F for L > 0 under suitable boundaries.
Symbols & assumptions
η labels phases or orientations; L is mobility. Physical time calibration requires kinetic information.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Take mobility, positive quadratic free-energy curvature, and gradient coefficient all equal to one, with initial cos x.
η(x,1)=e−2cosx
Solution. The local and gradient terms each contribute −A to the amplitude equation. Integrate A′ = −2A.
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Orange point: horizontal coordinate 3.1416, calculated vertical coordinate -0.13534. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 3.1416, into the displayed formula to obtain -0.13534 on the vertical axis. Values are rounded for display.
Scope. Linear quadratic-free-energy special case; domain walls of a double-well model are not represented.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
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Uses a periodic density-like field to represent crystalline ordering.
Micro / mesoPhysical model
Mathematical model & short derivation
Representative formulation
F=∫{21ψ[r+(q02+∇2)2]ψ+ψ4/4}dV∂tψ=M∇2(δF/δψ)
Derivation / construction sketch
Choose a free-energy operator favoring spatial modulation at wave number q₀.
Add a stabilizing nonlinear term.
Use conserved gradient flow for the density-like field ψ.
Symbols & assumptions
A common dimensionless phase-field-crystal form; r is a control parameter, not a spatial coordinate here.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Linearize ∂tψ = ∇²[(r+(1+∇²)²)ψ+ψ³] about ψ = 0 with r = 1; initial amplitude A₀ = 0.01 and wave number one.
δψ(x,1)=A0e−1cosx
Solution. The operator (1+∂xx) annihilates cos x. The remaining linear amplitude equation is A′ = −A.
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Orange point: horizontal coordinate 3.1416, calculated vertical coordinate -0.0036788. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 3.1416, into the displayed formula to obtain -0.0036788 on the vertical axis. Values are rounded for display.
Scope. Linearized small-perturbation solution; the cubic term is omitted.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
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Assign each lattice site a discrete grain-orientation label.
Penalize boundaries between unlike neighboring labels.
Propose label changes and accept energy-lowering or thermally weighted moves.
Symbols & assumptions
Monte Carlo steps are not physical time without calibration; lattice anisotropy and temperature choice affect kinetics.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. For a three-state two-site Potts pair with energy −J when the states agree, compute the equilibrium agreement probability.
Psame=eu+q−1eu,q=3
Solution. There are q agreeing states with Boltzmann weight exp(J/kBT), and q(q−1) disagreeing states with weight one. Normalize their sums.
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Orange point: horizontal coordinate 2.5, calculated vertical coordinate 0.85898. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.85898 on the vertical axis. Values are rounded for display.
Scope. Finite equilibrium toy problem, not a simulated grain-growth history.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
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Represent a dislocation by line segments with Burgers vector b and tangent ξ.
Compute the local stress from external loads and other defects.
Use the Peach–Koehler force with a mobility relation to evolve the line.
Symbols & assumptions
Force is per unit length; M may be a tensor and glide/climb constraints apply. Junction reactions need additional rules.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Example 1: Straight dislocation with constant mobility
Straight dislocation with constant mobility
Problem & parameters. Take one straight segment, constant force per length f = 1 N/m and mobility M = 1 m²/(N·s), starting at x = 0.
x(t)=Mft
Solution. The overdamped mobility law gives constant velocity Mf; integrate with the initial position.
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Orange point: horizontal coordinate 2.5, calculated vertical coordinate 2.5. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 2.5 on the vertical axis. Values are rounded for display.
Scope. Illustrative coefficients; interactions, pinning, and changing segment geometry are excluded.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
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Tracks the distribution of particle sizes or other internal properties.
Micro / mesoPhysical model
Mathematical model & short derivation
Representative formulation
∂t∂n+∇x⋅(un)+∂s∂(Gn)=B−D
Derivation / construction sketch
Count particles in a small spatial and size interval.
Balance transport in position x and growth in internal coordinate s.
Add births B and deaths D from nucleation, breakup or aggregation.
Symbols & assumptions
n is number density in size space, G the size-growth rate. Breakage and aggregation kernels close the model.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. For ∂tn+∂sn = 0 use n(s,0) = exp[−(s−2)²], constant growth G = 1, and compatible boundary inflow. Plot t = 1.
n(s,1)=e−(s−3)2
Solution. Along characteristics s−t is constant. Therefore n(s,t) = n₀(s−t).
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Orange point: horizontal coordinate 3, calculated vertical coordinate 1. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 3, into the displayed formula to obtain 1 on the vertical axis. Values are rounded for display.
Scope. Analytical solution or constitutive evaluation for the stated special case. It is not a general solution of the full model family.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
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Relates pressure, volume and temperature for a dilute noninteracting gas.
Cross-scalePhysical model
Mathematical model & short derivation
Representative formulation
pV=nRT
Derivation / construction sketch
Use kinetic theory for dilute, noninteracting particles.
Relate pressure to momentum transfer at the walls and translational energy to temperature.
With N = nNA and R = NAkB, obtain the ideal-gas relation.
Symbols & assumptions
n is amount in moles, V volume and T absolute temperature; intermolecular interactions and finite molecular volume are neglected.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Practical applications & products
Application area
Thermodynamics & equilibrium
Practical use
Estimating the amount of air in a low-pressure vessel.
Problem & parameters. Hold temperature and amount of ideal gas fixed while varying its volume.
pV∗/(nRT)=1/(V/V∗)
Solution. Solve pV = nRT for pressure and divide by the reference pressure nRT/V*.
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Orange point: horizontal coordinate 2.75, calculated vertical coordinate 0.36364. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 2.75, into the displayed formula to obtain 0.36364 on the vertical axis. Values are rounded for display.
Scope. Analytical solution or constitutive evaluation for the stated special case. It is not a general solution of the full model family.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
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Adds molecular attraction and excluded volume to an ideal gas model.
Cross-scalePhysical model
Mathematical model & short derivation
Representative formulation
(p+a/v2)(v−b)=RT
Derivation / construction sketch
Replace available molar volume v by v−b to represent excluded space.
Correct measured pressure by a/v² to account for attraction.
Apply the ideal-gas relation to the corrected variables.
Symbols & assumptions
v is molar volume; a and b are substance parameters. This is a qualitative equation of state near critical and coexistence regions.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Use the reduced van der Waals equation at T/Tc = 1.2.
p/pc=3v−18(1.2)−v23
Solution. Insert the critical scalings Vc = 3b, pc = a/(27b²), and Tc = 8a/(27Rb), then evaluate the reduced expression.
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Orange point: horizontal coordinate 2.3, calculated vertical coordinate 1.06. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 2.3, into the displayed formula to obtain 1.06 on the vertical axis. Values are rounded for display.
Scope. Analytical solution or constitutive evaluation for the stated special case. It is not a general solution of the full model family.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
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Uses a cubic equation of state for real-fluid behavior.
Cross-scalePhysical model
Mathematical model & short derivation
Representative formulation
p=v−bRT−v(v+b)+b(v−b)a(T)
Derivation / construction sketch
Begin with a repulsive excluded-volume term.
Choose a rational attraction term that yields a cubic equation in molar volume.
Fit critical-point constraints and a temperature-dependent attraction to obtain the Peng–Robinson parameters.
Symbols & assumptions
v is molar volume; a(T) and b require critical properties and an acentric-factor correlation.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. At fixed temperature choose aα/(RTb) = 2 and evaluate the Peng–Robinson pressure.
pb/(RT)=v−11−v2+2v−12
Solution. Divide its repulsive and attractive terms by RT/b and substitute v = Vₘ/b.
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Orange point: horizontal coordinate 3.75, calculated vertical coordinate 0.26637. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 3.75, into the displayed formula to obtain 0.26637 on the vertical axis. Values are rounded for display.
Scope. Illustrative EOS parameters; not a fitted fluid or a phase-equilibrium calculation.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
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Uses a temperature-dependent attraction correction in a cubic fluid model.
Cross-scalePhysical model
Mathematical model & short derivation
Representative formulation
p=v−bRT−v(v+b)aα(T)
Derivation / construction sketch
Retain the Redlich–Kwong cubic volume dependence.
Replace its temperature factor with a fitted α(T).
Determine coefficients from critical properties and vapor-pressure behavior.
Symbols & assumptions
Soave–Redlich–Kwong form; mixing rules are additionally required for mixtures.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. At fixed temperature choose aα/(RTb) = 2 for the SRK equation.
pb/(RT)=v−11−v(v+1)2
Solution. Divide the EOS by RT/b and evaluate both terms using the reduced molar volume.
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Orange point: horizontal coordinate 3.75, calculated vertical coordinate 0.25136. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 3.75, into the displayed formula to obtain 0.25136 on the vertical axis. Values are rounded for display.
Scope. Illustrative parameters; the temperature dependence of α is fixed for this isotherm.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
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Represents nonideal behavior as a density or pressure expansion.
Cross-scalePhysical model
Mathematical model & short derivation
Representative formulation
Z=RTpv=1+vB(T)+v2C(T)+⋯
Derivation / construction sketch
Expand the compressibility factor about zero molar density.
Group pair, triplet and higher interaction effects into virial coefficients.
Truncate only where omitted density powers are small.
Symbols & assumptions
B and C are temperature-dependent molar virial coefficients; this low-density expansion may converge poorly near condensation.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Use scaled second and third virial coefficients 0.2 and 0.05 over a dilute density interval.
Z=1+0.2ρ∗+0.05ρ∗2
Solution. Substitute the reduced density into Z = 1+Bρ+Cρ². At zero density it recovers the ideal-gas limit Z = 1.
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Orange point: horizontal coordinate 0.5, calculated vertical coordinate 1.1125. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 1.1125 on the vertical axis. Values are rounded for display.
Scope. Truncated low-density illustrative expansion, not an extrapolation to dense fluids.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
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Finds equilibrium by minimizing free energy under conservation constraints.
Cross-scalePhysical model
Mathematical model & short derivation
Representative formulation
minG=i∑niμiAn=b,ni≥0
Derivation / construction sketch
Choose species amounts n as unknowns and encode elemental conservation with A.
At fixed temperature and pressure, stable equilibrium minimizes Gibbs energy.
Stationarity along an allowed reaction gives Σᵢνᵢμᵢ = 0.
Symbols & assumptions
μᵢ are chemical potentials dependent on composition; metastability and missing phases can alter the solution found.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Take an ideal binary solution with equal pure-component reference energies. Find the composition dependence of its mixing free energy.
Δg/(RT)=xlnx+(1−x)ln(1−x)
Solution. Sum the two ideal mixing contributions. Differentiation gives ln[x/(1−x)] = 0, so the minimum is at x = 1/2.
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Orange point: horizontal coordinate 0.5, calculated vertical coordinate -0.69315. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain -0.69315 on the vertical axis. Values are rounded for display.
Scope. Ideal-solution Gibbs term; real CALPHAD databases include additional phase and interaction terms. Conserved bulk composition constrains accessible equilibria.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
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Combines assessed phase free energies to predict equilibria.
Cross-scalePhysical model
Mathematical model & short derivation
Representative formulation
Gp(x,T)=i∑xiGip(T)+RTi∑xilnxi+Gexcessp
Derivation / construction sketch
Assign a free-energy function to each candidate phase p.
Combine reference-state, ideal-mixing and assessed excess contributions.
Minimize total free energy subject to overall composition to construct phase equilibrium.
Symbols & assumptions
Representative substitutional-solution CALPHAD form; sublattice and magnetic models add terms and require assessed databases.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Take an ideal binary solution with equal pure-component reference energies. Find the composition dependence of its mixing free energy.
Δg/(RT)=xlnx+(1−x)ln(1−x)
Solution. Sum the two ideal mixing contributions. Differentiation gives ln[x/(1−x)] = 0, so the minimum is at x = 1/2.
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Orange point: horizontal coordinate 0.5, calculated vertical coordinate -0.69315. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain -0.69315 on the vertical axis. Values are rounded for display.
Scope. Ideal-solution Gibbs term; real CALPHAD databases include additional phase and interaction terms. Conserved bulk composition constrains accessible equilibria.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
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Assume neighbors around a molecule have a composition different from the bulk.
Weight local interactions with nonrandomness factors G.
Differentiate nGE/(RT) with respect to component amounts to obtain ln γᵢ.
Symbols & assumptions
GE is molar excess Gibbs energy; τ are dimensionless interaction parameters and α nonrandomness parameters.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Set NRTL interaction parameters to zero; for UNIQUAC also take identical molecular sizes and shapes with zero interaction energies.
a1=x1,γ1=1
Solution. Under these restrictions the excess contribution vanishes and γ₁ = 1; activity is γ₁x₁.
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Orange point: horizontal coordinate 0.5, calculated vertical coordinate 0.5. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 0.5 on the vertical axis. Values are rounded for display.
Scope. Ideal-mixture limiting case only; unequal molecular sizes in UNIQUAC can retain a combinatorial contribution.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
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Separate mixture nonideality into molecular size/shape effects and interaction-energy effects.
Represent those contributions using volume and surface fractions.
Take the partial-molar derivative to obtain activity coefficients.
Symbols & assumptions
UNIQUAC structure; volume parameters rᵢ, surface parameters qᵢ and binary interaction parameters are needed. Derivative holds T, p and other component amounts fixed.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Set NRTL interaction parameters to zero; for UNIQUAC also take identical molecular sizes and shapes with zero interaction energies.
a1=x1,γ1=1
Solution. Under these restrictions the excess contribution vanishes and γ₁ = 1; activity is γ₁x₁.
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Orange point: horizontal coordinate 0.5, calculated vertical coordinate 0.5. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 0.5 on the vertical axis. Values are rounded for display.
Scope. Ideal-mixture limiting case only; unequal molecular sizes in UNIQUAC can retain a combinatorial contribution.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
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Approximates ionic activity using screened electrostatic interactions.
Cross-scalePhysical model
Mathematical model & short derivation
Representative formulation
log10γi=−Azi2II=21j∑cjzj2
Derivation / construction sketch
Linearize the Poisson–Boltzmann equation for weak electrostatic potentials.
Solve for the screened potential surrounding an ion.
Use the resulting electrostatic free-energy correction to obtain the limiting activity law.
Symbols & assumptions
Dilute-solution limiting law; concentrations, standard state and coefficient A must use a consistent convention.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. For a monovalent ion in water near 25 °C use the Debye–Hückel limiting-law coefficient A = 0.509 (mol/L)⁻¹ᐟ².
log10γ=−0.509I
Solution. Set charge magnitude to one in log₁₀γ = −Az²√I. Evaluate only at dilute ionic strengths.
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Orange point: horizontal coordinate 0.005, calculated vertical coordinate -0.035992. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.005, into the displayed formula to obtain -0.035992 on the vertical axis. Values are rounded for display.
Scope. Limiting-law illustration; specific ion interactions and concentrated solutions are excluded.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
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Relates reaction rates to species concentrations and reaction orders.
Cross-scalePhysical model
Mathematical model & short derivation
Representative formulation
r=kfi∏ciαi−kri∏ciβic˙i=νir
Derivation / construction sketch
Represent the frequency of elementary forward and reverse reactions by reactant encounters.
Subtract reverse from forward progress rates.
Multiply net progress by each species’ stoichiometric change to obtain its source.
Symbols & assumptions
For elementary reactions, exponents follow reactant stoichiometry; empirical overall reactions can have different orders.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. For a single irreversible first-order reaction A → products in a constant-volume batch, use τ = kt and y = cA/cA0.
y(τ)=e−τ
Solution. Separate dy/dτ = −y, integrate ln(y) = −τ + C, and apply y(0) = 1.
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Orange point: horizontal coordinate 2.5, calculated vertical coordinate 0.082085. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.082085 on the vertical axis. Values are rounded for display.
Scope. Exact one-mode reduction with constant coefficients; additional coupled physics is excluded.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
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Relates a rate coefficient to temperature through an activation energy.
Cross-scalePhysical model
Mathematical model & short derivation
Representative formulation
k(T)=Aexp[−Ea/(RT)]
Derivation / construction sketch
Approximate the fraction of thermal configurations able to cross a barrier by a Boltzmann factor.
Multiply that fraction by an effective attempt-frequency factor.
Taking ln k yields ln A − Ea/(RT), which motivates an Arrhenius plot.
Symbols & assumptions
Ea is molar activation energy; A and Ea may vary over broad temperature ranges.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Example 1: Temperature dependence of an activated rate
Temperature dependence of an activated rate
Problem & parameters. Hold activation energy Ea > 0 and prefactor A constant.
k/A=e−1/θ,θ=RT/Ea
Solution. Insert the scaled temperature into k = A exp(−Ea/RT).
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Orange point: horizontal coordinate 0.55, calculated vertical coordinate 0.16232. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.55, into the displayed formula to obtain 0.16232 on the vertical axis. Values are rounded for display.
Scope. Analytical solution or constitutive evaluation for the stated special case. It is not a general solution of the full model family.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
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Estimates reaction rates from a free-energy barrier.
Cross-scalePhysical model
Mathematical model & short derivation
Representative formulation
k=κhkBTexp[−ΔG‡/(RT)]
Derivation / construction sketch
Assume reactants are in quasi-equilibrium with configurations at a dividing surface.
Convert their statistical population into a crossing flux.
Multiply by a transmission coefficient κ to account for recrossing or other corrections.
Symbols & assumptions
ΔG‡ is a molar activation free energy consistent with the standard state; h is Planck’s constant.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Example 1: Transition-state rate at fixed activation free energy
Transition-state rate at fixed activation free energy
Problem & parameters. Take transmission coefficient one and treat the molar activation free energy as constant over the displayed interval.
kh/(kBT)=e−1/θ,θ=RT/ΔG‡
Solution. Divide the Eyring expression by its kBT/h prefactor and substitute the scaled temperature.
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Orange point: horizontal coordinate 0.55, calculated vertical coordinate 0.16232. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.55, into the displayed formula to obtain 0.16232 on the vertical axis. Values are rounded for display.
Scope. Illustrative fixed-barrier curve; real activation free energy can vary with temperature.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
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Approximates enzyme reaction rates with substrate saturation.
Cross-scalePhysical model
Mathematical model & short derivation
Representative formulation
v=KM+[S]Vmax[S]KM=k1k−1+kcat
Derivation / construction sketch
Use E+S ⇌ ES → E+P.
Apply the quasi-steady-state condition to ES and enzyme conservation [E]T = [E]+[ES].
Solve for [ES] and substitute into v = kcat[ES].
Symbols & assumptions
Initial-rate, simple single-substrate model; Vmax = kcat[E]T. Substrate depletion, reversibility and inhibition need extensions.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. For Michaelis–Menten set x = substrate/Km and y = v/Vmax. For Langmuir adsorption set x = KP and y = occupied-site fraction.
y=1+xx
Solution. Solve the binding or adsorption balance to give occupied fraction x/(1+x). The half-saturation point is x = 1.
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Orange point: horizontal coordinate 4, calculated vertical coordinate 0.8. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 4, into the displayed formula to obtain 0.8 on the vertical axis. Values are rounded for display.
Scope. Single-substrate steady enzyme law or single-species equilibrium adsorption, as appropriate to the entry.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Models adsorption on equivalent sites with finite occupancy.
Cross-scalePhysical model
Mathematical model & short derivation
Representative formulation
θ=1+KPKP
Derivation / construction sketch
Balance adsorption kaP(1−θ) against desorption kdθ.
Set the net rate to zero at equilibrium.
Solve for occupied-site fraction θ with K = ka/kd.
Symbols & assumptions
Equivalent independent sites, monolayer adsorption and gas pressure P; concentration can replace pressure with a compatible equilibrium constant.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. For Michaelis–Menten set x = substrate/Km and y = v/Vmax. For Langmuir adsorption set x = KP and y = occupied-site fraction.
y=1+xx
Solution. Solve the binding or adsorption balance to give occupied fraction x/(1+x). The half-saturation point is x = 1.
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Orange point: horizontal coordinate 4, calculated vertical coordinate 0.8. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 4, into the displayed formula to obtain 0.8 on the vertical axis. Values are rounded for display.
Scope. Single-substrate steady enzyme law or single-species equilibrium adsorption, as appropriate to the entry.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Assume both reactants adsorb competitively on equivalent sites.
Use Langmuir expressions for coverages θA and θB.
For a rate-limiting reaction between adsorbates, set r = kθAθB.
Symbols & assumptions
One representative Langmuir–Hinshelwood mechanism; different adsorption or rate-limiting steps produce different denominators.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Example 1: Competing adsorption and surface reaction
Competing adsorption and surface reaction
Problem & parameters. Use the illustrative Langmuir–Hinshelwood rate r/r* = x/(1+x)², with other factors held constant.
r/r∗=(1+x)2x
Solution. Differentiate: the slope is (1−x)/(1+x)³. Thus the rate peaks at x = 1 and decreases under strong site blocking.
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Orange point: horizontal coordinate 4, calculated vertical coordinate 0.16. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 4, into the displayed formula to obtain 0.16 on the vertical axis. Values are rounded for display.
Scope. One specified adsorption-limited rate law; the family contains many different mechanisms.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Relates diffusive flux to concentration gradients.
Cross-scalePhysical model
Mathematical model & short derivation
Representative formulation
J=−D∇c∂t∂c=∇⋅(D∇c)
Derivation / construction sketch
Approximate diffusive flux as linear in a small concentration gradient.
Combine that constitutive relation with species conservation ∂tc+∇·J = 0.
For constant D, this reduces to ∂tc = D∇²c.
Symbols & assumptions
Fickian diffusion with no advection or reactions; D may be anisotropic or concentration-dependent.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Solve ∂τu = ∂ξξu with u(0,τ)=u(1,τ)=0 and initial sin(πξ), then plot τ = 0.1.
u(ξ,τ)=sin(πξ)e−π2τ,τ=0.1
Solution. The sine satisfies both zero end values. Its second derivative is −π² times itself; the amplitude solves a′ = −π²a.
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Orange point: horizontal coordinate 0.5, calculated vertical coordinate 0.37271. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 0.37271 on the vertical axis. Values are rounded for display.
Scope. Fickian constant-diffusivity slab. Maxwell–Stefan reduces to this form for an ideal binary mixture with constant total concentration and diffusivity.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
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Represents multicomponent diffusion through interspecies friction.
Cross-scalePhysical model
Mathematical model & short derivation
Representative formulation
−∇xi=j=i∑cDijxjNi−xiNj
Derivation / construction sketch
Balance thermodynamic driving forces against pairwise interspecies friction.
For an ideal isothermal isobaric mixture, use mole-fraction gradients as the driving terms.
Express relative velocities through molar fluxes to obtain coupled diffusion equations.
Symbols & assumptions
xᵢ are mole fractions, Nᵢ molar fluxes, c total molar concentration and Dᵢⱼ binary diffusivities; nonideal or nonisobaric cases add terms.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Solve ∂τu = ∂ξξu with u(0,τ)=u(1,τ)=0 and initial sin(πξ), then plot τ = 0.1.
u(ξ,τ)=sin(πξ)e−π2τ,τ=0.1
Solution. The sine satisfies both zero end values. Its second derivative is −π² times itself; the amplitude solves a′ = −π²a.
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Orange point: horizontal coordinate 0.5, calculated vertical coordinate 0.37271. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 0.37271 on the vertical axis. Values are rounded for display.
Scope. Fickian constant-diffusivity slab. Maxwell–Stefan reduces to this form for an ideal binary mixture with constant total concentration and diffusivity.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Combines bulk transport, diffusion and reaction sources.
Cross-scalePhysical model
Mathematical model & short derivation
Representative formulation
∂t∂c+∇⋅(uc)=∇⋅(D∇c)+R(c)
Derivation / construction sketch
Write local conservation with total flux uc+J.
Insert Fickian diffusive flux J = −D∇c.
Add the net production rate R from reactions.
Symbols & assumptions
u is carrier velocity; compressibility, variable porosity and multiple species may require modified storage and transport terms.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. On the infinite line solve ut+ux = 0.1uxx−0.2u with u(x,0)=exp(−x²). Plot t = 1.
u(x,1)=1.4e−0.2exp[−(x−1)2/1.4]
Solution. Advection translates the center by t. Diffusion increases the Gaussian width from 1 to 1+0.4t; first-order loss multiplies its conserved-mass diffusion solution by exp(−0.2t).
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Orange point: horizontal coordinate 1, calculated vertical coordinate 0.69195. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 1, into the displayed formula to obtain 0.69195 on the vertical axis. Values are rounded for display.
Scope. Analytical solution or constitutive evaluation for the stated special case. It is not a general solution of the full model family.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
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Assumes a well-mixed reactor with inlet and outlet flows.
Cross-scalePhysical model
Mathematical model & short derivation
Representative formulation
Vdtdc=Q(cin−c)+VR(c)
Derivation / construction sketch
Apply a species balance to the reactor volume.
Assume perfect mixing, so outlet concentration equals reactor concentration.
For equal inlet/outlet volumetric flow Q and constant V, collect flow and reaction terms.
Symbols & assumptions
Uniform composition and temperature assumed unless an energy balance is added; residence time is V/Q.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. At steady state a well-mixed reactor consumes A by a first-order reaction at rate kcA.
cout/cin=1/(1+Da)
Solution. Balance Qcin−Qcout−kVcout = 0 and solve for cout.
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Orange point: horizontal coordinate 2.5, calculated vertical coordinate 0.28571. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.28571 on the vertical axis. Values are rounded for display.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
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Approximates axial evolution without axial back-mixing.
Cross-scalePhysical model
Mathematical model & short derivation
Representative formulation
QdVdc=R(c)
Derivation / construction sketch
Apply steady species conservation to a thin reactor slice.
Assume negligible axial diffusion and uniform properties across each section.
Divide the flow change by slice volume and take the differential limit.
Symbols & assumptions
Constant volumetric flow Q shown; gas expansion or varying density requires a molar-flow formulation.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. For an isothermal PFR with constant velocity u and first-order consumption k, use τ = kz/u and y = c/cin.
y(τ)=e−τ
Solution. Separate dy/dτ = −y, integrate ln(y) = −τ + C, and apply y(0) = 1.
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Orange point: horizontal coordinate 2.5, calculated vertical coordinate 0.082085. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.082085 on the vertical axis. Values are rounded for display.
Scope. Exact axial concentration profile in ideal plug flow; the horizontal coordinate is residence time kz/u, not laboratory time.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
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Evolves composition and energy in a closed reacting charge.
Cross-scalePhysical model
Mathematical model & short derivation
Representative formulation
dtdci=Ri(c,T)
Derivation / construction sketch
Apply species conservation to a closed, well-mixed vessel.
Set inlet and outlet flows to zero.
For constant volume, divide the species production rate by vessel volume.
Symbols & assumptions
An energy balance determines T if the batch is not isothermal; variable volume changes the concentration equation.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. For a single irreversible first-order reaction A → products in a constant-volume batch, use τ = kt and y = cA/cA0.
y(τ)=e−τ
Solution. Separate dy/dτ = −y, integrate ln(y) = −τ + C, and apply y(0) = 1.
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Orange point: horizontal coordinate 2.5, calculated vertical coordinate 0.082085. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.082085 on the vertical axis. Values are rounded for display.
Scope. Exact one-mode reduction with constant coefficients; additional coupled physics is excluded.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
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Conserves mass and momentum for a viscous continuum fluid.
Continuum / componentPhysical model
Mathematical model & short derivation
Representative formulation
ρ(∂tu+u⋅∇u)=−∇p+μ∇2u+ρg∇⋅u=0
Derivation / construction sketch
Apply conservation of momentum to a fluid element.
Split stress into pressure and Newtonian viscous stress.
For constant density and viscosity, substitute this stress into the balance to obtain incompressible Navier–Stokes.
Symbols & assumptions
u is velocity, p pressure, ρ density and μ dynamic viscosity. Compressible flow also needs energy and an equation of state.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Take steady, fully developed incompressible flow with constant viscosity between fixed parallel plates. For Hagen–Poiseuille use the equivalent diameter cut through a round pipe.
u/Umax=1−ξ2
Solution. The axial momentum equation becomes a constant second derivative. Integrate twice and impose no slip at both walls to obtain a parabola.
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Orange point: horizontal coordinate 0, calculated vertical coordinate 1. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 1 on the vertical axis. Values are rounded for display.
Scope. Exact laminar benchmark. Plate and pipe pressure-to-maximum-speed factors differ; the plotted normalized profile is identical. DNS here resolves this simple laminar case.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
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Neglects viscous stresses in compressible or incompressible flow.
Continuum / componentPhysical model
Mathematical model & short derivation
Representative formulation
ρDtDu=−∇p+ρg∂tρ+∇⋅(ρu)=0
Derivation / construction sketch
Use continuum mass and momentum balances.
Neglect viscous stresses while retaining pressure forces.
Combine with energy conservation and an equation of state when density varies.
Symbols & assumptions
D/Dt = ∂t+u·∇ is the material derivative. Inviscid approximations do not reproduce no-slip wall layers.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Linearize inviscid Euler flow about a uniform rest state and use a sinusoidal pressure perturbation.
u(ξ,0)=sin(2πξ)
Solution. A sinusoidal traveling-wave solution is u = sin[2π(ξ−τ)]. Set τ = 0 to obtain the plotted snapshot.
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Orange point: horizontal coordinate 0.5, calculated vertical coordinate 1.2246e-16. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 1.2246e-16 on the vertical axis. Values are rounded for display.
Scope. Linear acoustic limit of Euler flow, not a finite-amplitude compressible flow solution.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
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Scale momentum transport using a characteristic length L and velocity U.
When Reynolds number ρUL/μ is much less than one, inertia is small.
Drop inertial terms from the incompressible Navier–Stokes equation.
Symbols & assumptions
Steady creeping-flow form; rapid transients can require unsteady inertia even when convective inertia is small.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Take steady, fully developed incompressible flow with constant viscosity between fixed parallel plates. For Hagen–Poiseuille use the equivalent diameter cut through a round pipe.
u/Umax=1−ξ2
Solution. The axial momentum equation becomes a constant second derivative. Integrate twice and impose no slip at both walls to obtain a parabola.
Open this section to load its graph, or use the full-size example link below.
Orange point: horizontal coordinate 0, calculated vertical coordinate 1. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 1 on the vertical axis. Values are rounded for display.
Scope. Exact laminar benchmark. Plate and pipe pressure-to-maximum-speed factors differ; the plotted normalized profile is identical. DNS here resolves this simple laminar case.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Represents irrotational velocity using a scalar potential.
Continuum / componentPhysical model
Mathematical model & short derivation
Representative formulation
u=∇ϕ∇2ϕ=0
Derivation / construction sketch
Assume irrotational velocity, ∇×u = 0, in a suitable simply connected region.
Introduce a velocity potential φ.
Substitute into incompressible continuity ∇·u = 0 to obtain Laplace’s equation.
Symbols & assumptions
Incompressible potential flow; circulation or compressibility requires additional treatment. It cannot directly predict viscous drag.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Find surface pressure for incompressible, inviscid, irrotational uniform flow around a circular cylinder without circulation.
Cp=1−4sin2θ
Solution. Potential flow gives surface speed 2U∞ sin θ. Bernoulli’s equation then gives Cp = 1−(u/U∞)².
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Orange point: horizontal coordinate 3.1416, calculated vertical coordinate 1. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 3.1416, into the displayed formula to obtain 1 on the vertical axis. Values are rounded for display.
Scope. No viscosity or separation; this ideal model does not predict real cylinder drag.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
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Resolves thin near-wall regions with scale-based simplifications.
Continuum / componentPhysical model
Mathematical model & short derivation
Representative formulation
u∂xu+v∂yu=UedxdUe+ν∂yyu∂xu+∂yv=0
Derivation / construction sketch
Assume a thin steady two-dimensional layer near a wall.
Use its small thickness to neglect streamwise viscous diffusion relative to wall-normal diffusion.
Match pressure to the outer inviscid flow, giving the Uₑ pressure-gradient term.
Symbols & assumptions
u and v are tangential and normal velocities; Uₑ is external speed and ν kinematic viscosity. Separation challenges the simplest approximation.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. A flat wall suddenly moves at speed U beneath an initially stationary semi-infinite viscous fluid.
u/U=erfc(η),η=y/(2νt)
Solution. With no streamwise variation, momentum reduces to diffusion. Similarity substitution and the wall/far-field conditions give the complementary error function.
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Orange point: horizontal coordinate 1.5, calculated vertical coordinate 0.033895. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 1.5, into the displayed formula to obtain 0.033895 on the vertical axis. Values are rounded for display.
Scope. Stokes’ first problem, an unsteady boundary-layer benchmark; not the Blasius spatially developing solution.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
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Integrate μ∂yyu = ∂xp across the gap using no-slip conditions.
Integrate velocity to obtain flow rate and insert it into gap-volume conservation.
Symbols & assumptions
One-dimensional incompressible lubrication equation with one wall moving at U; h is gap thickness. Other wall motions change the Couette term.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Take steady, fully developed incompressible flow with constant viscosity between fixed parallel plates. For Hagen–Poiseuille use the equivalent diameter cut through a round pipe.
u/Umax=1−ξ2
Solution. The axial momentum equation becomes a constant second derivative. Integrate twice and impose no slip at both walls to obtain a parabola.
Open this section to load its graph, or use the full-size example link below.
Orange point: horizontal coordinate 0, calculated vertical coordinate 1. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 1 on the vertical axis. Values are rounded for display.
Scope. Exact laminar benchmark. Plate and pipe pressure-to-maximum-speed factors differ; the plotted normalized profile is identical. DNS here resolves this simple laminar case.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Predicts fully developed laminar flow in a circular pipe.
Continuum / componentPhysical model
Mathematical model & short derivation
Representative formulation
u(r)=4μLΔp(R2−r2)Q=8μLπR4Δp
Derivation / construction sketch
Assume steady, fully developed axisymmetric flow in a circular tube.
Integrate the axial viscous momentum equation with finite velocity gradient at r=0 and no slip at r=R.
Integrate the parabolic profile over the cross section.
Symbols & assumptions
Newtonian laminar flow in a tube of length L and radius R; entrance and non-Newtonian effects are omitted.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Take steady, fully developed incompressible flow with constant viscosity between fixed parallel plates. For Hagen–Poiseuille use the equivalent diameter cut through a round pipe.
u/Umax=1−ξ2
Solution. The axial momentum equation becomes a constant second derivative. Integrate twice and impose no slip at both walls to obtain a parabola.
Open this section to load its graph, or use the full-size example link below.
Orange point: horizontal coordinate 0, calculated vertical coordinate 1. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 1 on the vertical axis. Values are rounded for display.
Scope. Exact laminar benchmark. Plate and pipe pressure-to-maximum-speed factors differ; the plotted normalized profile is identical. DNS here resolves this simple laminar case.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
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Relates pipe pressure loss to friction factor and flow speed.
Continuum / componentPhysical model
Mathematical model & short derivation
Representative formulation
Δp=fDDL2ρU2
Derivation / construction sketch
Balance wall shear force τwπDL against pressure force ΔpπD²/4.
Define the Darcy friction factor fD = 8τw/(ρU²).
Substitute this definition into the force balance.
Symbols & assumptions
fD depends on Reynolds number and roughness. Do not confuse it with the Fanning friction factor, which is four times smaller.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Hold the Darcy friction factor f, pipe geometry, and density fixed.
Δp/(fLρU∗2/2D)=(U/U∗)2
Solution. Insert the mean speed into Darcy–Weisbach Δp = f(L/D)ρU²/2.
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Orange point: horizontal coordinate 1.5, calculated vertical coordinate 2.25. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 1.5, into the displayed formula to obtain 2.25 on the vertical axis. Values are rounded for display.
Scope. Fixed-friction-factor illustration; f usually varies with Reynolds number and roughness.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
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Represent shear stress versus shear rate by a fitted power law.
Divide stress by shear rate to define apparent viscosity.
For n<1 the apparent viscosity decreases as shear rate increases.
Symbols & assumptions
Simple positive-shear-rate form; K is consistency and n the flow index. Low- and high-rate viscosity plateaus are not represented.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Choose positive shear rates and power-law exponent n = 1/2, with reference stress K√(reference rate).
τ/τ∗=(γ˙/γ˙∗)1/2
Solution. Substitute n = 1/2 into τ = Kγ̇ⁿ.
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Orange point: horizontal coordinate 2, calculated vertical coordinate 1.4142. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 2, into the displayed formula to obtain 1.4142 on the vertical axis. Values are rounded for display.
Scope. Steady shear constitutive evaluation; no low- or high-shear viscosity plateau is included.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
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Represents a material with a yield stress and post-yield viscosity.
Continuum / componentPhysical model
Mathematical model & short derivation
Representative formulation
γ˙=0for ∣τ∣≤τyτ=τysign(γ˙)+μpγ˙otherwise
Derivation / construction sketch
Assume a rigid response below a yield stress.
Above yield, add a constant yield contribution to a linear viscous stress.
Invert the relation to obtain shear rate for a specified stress.
Symbols & assumptions
τy is yield stress, μp plastic viscosity; numerical regularization changes the ideal unyielded region.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Increase a nonnegative applied shear stress on an ideal Bingham material.
μpγ˙/τy=max(s−1,0),s=τ/τy
Solution. Below yield, the shear rate is zero. Above yield, solve τ = τy+μpγ̇ for the rate.
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Orange point: horizontal coordinate 1.5, calculated vertical coordinate 0.5. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 1.5, into the displayed formula to obtain 0.5 on the vertical axis. Values are rounded for display.
Scope. Analytical solution or constitutive evaluation for the stated special case. It is not a general solution of the full model family.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
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Combines yield stress with nonlinear post-yield flow.
Continuum / componentPhysical model
Mathematical model & short derivation
Representative formulation
τ=τysign(γ˙)+K∣γ˙∣nsign(γ˙)when flowing
Derivation / construction sketch
Start with the Bingham yield condition.
Replace the post-yield linear viscous term by a power-law term.
Set shear rate to zero for stresses below the yield threshold.
Symbols & assumptions
τy, K and n must be fitted for the material and temperature; this is a simple-shear representation.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Use exponent n = 1/2 and define g so that Kγ̇ⁿ/τy = √g. Evaluate the yielded branch.
τ/τy=1+g1/2
Solution. Insert the chosen exponent into τ = τy+Kγ̇ⁿ.
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Orange point: horizontal coordinate 2, calculated vertical coordinate 2.4142. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 2, into the displayed formula to obtain 2.4142 on the vertical axis. Values are rounded for display.
Scope. Positive yielded branch only; at zero rate the unyielded model allows a range of stresses.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
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Combines solvent viscosity with an elastic polymer stress.
Continuum / componentPhysical model
Mathematical model & short derivation
Representative formulation
τp+λτ∇p=2ηpDσ=−pI+2ηsD+τp
Derivation / construction sketch
Represent polymer relaxation with a Maxwell-like stress evolution.
Replace an ordinary time derivative by an upper-convected derivative to preserve frame invariance.
Add a Newtonian solvent contribution.
Symbols & assumptions
D = (∇u+∇uᵀ)/2; τp∇ = ∂tτp+u·∇τp−(∇u)τp−τp(∇u)ᵀ. λ is relaxation time.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. After a small deformation, hold the fluid motionless. A homogeneous Oldroyd-B polymer shear stress obeys λdτp/dt+τp=0.
y(τ)=e−τ
Solution. Separate dy/dτ = −y, integrate ln(y) = −τ + C, and apply y(0) = 1.
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Orange point: horizontal coordinate 2.5, calculated vertical coordinate 0.082085. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.082085 on the vertical axis. Values are rounded for display.
Scope. Zero-velocity, homogeneous stress-relaxation subproblem; convected terms vanish and the solvent stress is zero.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
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Models mean flow with closure for unresolved turbulent stresses.
Continuum / componentPhysical model
Mathematical model & short derivation
Representative formulation
ρ(∂tU+U⋅∇U)=−∇P+μ∇2U−ρ∇⋅⟨u′u′⟩
Derivation / construction sketch
Decompose velocity into a mean U and fluctuation u′.
Average the nonlinear momentum equation.
The product of fluctuations produces Reynolds stress, requiring a closure model.
Symbols & assumptions
Constant-density form; the averaging operation and boundary conditions must be consistent.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Verify the molecular-viscosity momentum equation using fully developed plane Poiseuille flow with turbulent or subgrid stresses disabled.
u/Umax=1−ξ2
Solution. A constant pressure gradient gives μu″ = dp/dx. Apply no slip at the two walls and normalize by the center speed.
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Orange point: horizontal coordinate 0, calculated vertical coordinate 1. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 1 on the vertical axis. Values are rounded for display.
Scope. Laminar-limit verification only; it neither models turbulence nor validates a RANS, LES, or DES closure.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
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Uses a transported turbulence variable to obtain eddy viscosity.
Continuum / componentPhysical model
Mathematical model & short derivation
Representative formulation
νt=ν~fv1DtDν~=Pν~+Dν~−Wν~
Derivation / construction sketch
Introduce one transported modified eddy-viscosity variable ν̃.
Balance its modeled production, diffusion and near-wall destruction.
Convert ν̃ to physical eddy viscosity using the damping function fv1.
Symbols & assumptions
Schematic Spalart–Allmaras transport structure; full published functions, constants and wall-distance treatment are required for computation.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. For nonnegative working variable χ evaluate the standard SA eddy-viscosity mapping with cv1 = 7.1.
νt/ν=χχ3+7.13χ3
Solution. Compute the damping function fv1 = χ³/(χ³+cv1³), then multiply by χ.
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Orange point: horizontal coordinate 10, calculated vertical coordinate 7.3643. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 10, into the displayed formula to obtain 7.3643 on the vertical axis. Values are rounded for display.
Scope. Algebraic closure contribution only, not a solution of the SA transport equation.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
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Uses turbulent kinetic energy and dissipation rate to close mean flow.
Continuum / componentPhysical model
Mathematical model & short derivation
Representative formulation
νt=Cμεk2DtDk=Pk−ε+diffusion
Derivation / construction sketch
Use k as turbulent kinetic energy and ε as its dissipation rate.
Dimensional analysis gives a turbulent viscosity scale k²/ε.
Close mean stresses with this viscosity and solve modeled transport equations for both k and ε.
Symbols & assumptions
The ε equation and boundary functions are essential; standard, RNG and realizable variants differ.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Hold dissipation ε = ε* fixed and use Cμ = 0.09.
νtϵ∗/k∗2=0.09(k/k∗)2
Solution. Substitute k into νt = Cμk²/ε.
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Orange point: horizontal coordinate 2, calculated vertical coordinate 0.36. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 2, into the displayed formula to obtain 0.36 on the vertical axis. Values are rounded for display.
Scope. Closure evaluation, not a prediction of k or ε from their coupled transport equations.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
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Uses turbulent kinetic energy and specific dissipation rate.
Continuum / componentPhysical model
Mathematical model & short derivation
Representative formulation
νt=ωkDtDk=Pk−β∗kω+diffusion
Derivation / construction sketch
Use a turbulence time scale proportional to 1/ω.
Multiply that time scale by kinetic energy k to form eddy viscosity.
Transport k and ω with modeled production, dissipation and diffusion.
Symbols & assumptions
Representative k–ω structure; β* and all other coefficients depend on the model version.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Hold specific dissipation ω = ω* > 0 and use the basic νt = k/ω relation.
νtω∗/k∗=k/k∗
Solution. Divide the closure by the reference viscosity k*/ω*.
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Orange point: horizontal coordinate 2, calculated vertical coordinate 2. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 2, into the displayed formula to obtain 2 on the vertical axis. Values are rounded for display.
Scope. Basic algebraic closure with fixed ω; model variants may include limiters.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
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Blends near-wall and outer-flow behavior with a shear-stress limiter.
Continuum / componentPhysical model
Mathematical model & short derivation
Representative formulation
νt=max(a1ω,SF2)a1k
Derivation / construction sketch
Blend k–ω behavior near walls with transformed k–ε behavior away from walls.
Limit the eddy viscosity using strain rate S and blending function F₂.
This restricts excessive turbulent shear stress in adverse pressure gradients.
Symbols & assumptions
SST models require two transport equations and blending functions; variants use different production limiting and strain/vorticity definitions.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Hold positive k and ω fixed. Evaluate νt = a1k/max(a1ω,SF2) with a1 = 0.31.
νtω/k=max(0.31,s)0.31,s=SF2/ω
Solution. Divide denominator and numerator by ω to expose the limiter transition at s = a1.
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Orange point: horizontal coordinate 1, calculated vertical coordinate 0.31. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 1, into the displayed formula to obtain 0.31 on the vertical axis. Values are rounded for display.
Scope. Algebraic SST limiter illustration; blending functions and transport equations are not solved.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. For a homogeneous Reynolds-stress anisotropy component use the reduced closure db/dt = −b/T with constant T.
y(τ)=e−τ
Solution. Separate dy/dτ = −y, integrate ln(y) = −τ + C, and apply y(0) = 1.
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Orange point: horizontal coordinate 2.5, calculated vertical coordinate 0.082085. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.082085 on the vertical axis. Values are rounded for display.
Scope. Isolated linear return-to-isotropy term; production, transport, and changing dissipation are excluded.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
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Resolves larger turbulent motions and models subgrid effects.
Continuum / componentPhysical model
Mathematical model & short derivation
Representative formulation
τijSGS=uiuj−uˉiuˉj
Derivation / construction sketch
Spatially filter the Navier–Stokes equations.
Filtering the nonlinear product differs from multiplying filtered velocities.
Represent that difference as subgrid stress and close it while resolving larger motions.
Symbols & assumptions
Filter width, mesh and numerical dissipation jointly determine the effective LES resolution.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Verify the molecular-viscosity momentum equation using fully developed plane Poiseuille flow with turbulent or subgrid stresses disabled.
u/Umax=1−ξ2
Solution. A constant pressure gradient gives μu″ = dp/dx. Apply no slip at the two walls and normalize by the center speed.
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Orange point: horizontal coordinate 0, calculated vertical coordinate 1. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 1 on the vertical axis. Values are rounded for display.
Scope. Laminar-limit verification only; it neither models turbulence nor validates a RANS, LES, or DES closure.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
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Relates subgrid eddy viscosity to resolved strain and filter scale.
Continuum / componentPhysical model
Mathematical model & short derivation
Representative formulation
νSGS=(CsΔ)22SˉijSˉij
Derivation / construction sketch
Assume a subgrid mixing length proportional to filter width Δ.
Use resolved strain to estimate an inverse time scale.
Multiply squared mixing length by that rate to obtain an eddy viscosity.
Symbols & assumptions
Cs is a coefficient and S̄ resolved strain. Wall damping or dynamic procedures may be needed.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Use Cs = 0.1 and constant filter width Δ.
νt/(Δ2S∗)=0.01(∣S∣/S∗)
Solution. Evaluate νt = (CsΔ)²|S|.
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Orange point: horizontal coordinate 2.5, calculated vertical coordinate 0.025. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.025 on the vertical axis. Values are rounded for display.
Scope. Constant-coefficient closure; no dynamic procedure or wall damping is included.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
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Combines RANS near walls with LES-like treatment away from them.
Continuum / componentPhysical model
Mathematical model & short derivation
Representative formulation
ℓDES=min(d,CDESΔ)
Derivation / construction sketch
Begin with a wall-distance-based RANS length scale d.
Replace it by a grid-related scale when that becomes smaller.
This enables LES-like behavior away from walls while keeping near-wall RANS treatment.
Symbols & assumptions
Representative original DES switch; delayed and improved delayed DES use shielding and other refinements.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Verify the molecular-viscosity momentum equation using fully developed plane Poiseuille flow with turbulent or subgrid stresses disabled.
u/Umax=1−ξ2
Solution. A constant pressure gradient gives μu″ = dp/dx. Apply no slip at the two walls and normalize by the center speed.
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Orange point: horizontal coordinate 0, calculated vertical coordinate 1. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 1 on the vertical axis. Values are rounded for display.
Scope. Laminar-limit verification only; it neither models turbulence nor validates a RANS, LES, or DES closure.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Tracks phase volume fractions to represent an interface.
Continuum / componentPhysical model
Mathematical model & short derivation
Representative formulation
∂tα+∇⋅(αu)=0ρ=αρ1+(1−α)ρ2
Derivation / construction sketch
Track the fraction α of one incompressible phase inside each cell.
Conserve that phase volume during advection.
Use the local fraction to combine material properties and reconstruct the interface.
Symbols & assumptions
Shared-velocity two-phase form without phase change; surface tension and interface compression are additional terms or numerical treatments.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Advect the initial smoothed interface α(x,0) = [1−tanh(5x)]/2 at unit velocity with no compression term.
α(x,1)=21[1−tanh(5(x−1))]
Solution. Characteristics give α(x,t) = α₀(x−t); evaluate t = 1.
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Orange point: horizontal coordinate 1, calculated vertical coordinate 0.5. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 1, into the displayed formula to obtain 0.5 on the vertical axis. Values are rounded for display.
Scope. Exact scalar-advection benchmark with a deliberately smooth interface; interface reconstruction and multiphase momentum are not solved.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
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Treats phases as interpenetrating continua with exchange terms.
Continuum / componentPhysical model
Mathematical model & short derivation
Representative formulation
∂t(αkρk)+∇⋅(αkρkuk)=Γkk∑αk=1
Derivation / construction sketch
Volume-average conservation separately for each phase.
Weight storage and fluxes by phase volume fraction αk.
Add interphase mass and momentum exchanges, then solve coupled phase equations.
Symbols & assumptions
A momentum equation is needed for each phase; drag, lift and other exchange forces require closures.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. For two homogeneous phases coupled only by linear interphase drag, scale time by the combined drag relaxation time and slip by its initial value.
y(τ)=e−τ
Solution. Separate dy/dτ = −y, integrate ln(y) = −τ + C, and apply y(0) = 1.
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Orange point: horizontal coordinate 2.5, calculated vertical coordinate 0.082085. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.082085 on the vertical axis. Values are rounded for display.
Scope. Subtract the two phase momentum balances to obtain a decaying relative velocity; spatial transport, pressure gradients, and phase change are absent.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
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Apply Newton’s law to an individual particle or droplet.
Represent fluid interaction by drag and, here, a buoyancy-corrected gravitational term.
Integrate particle velocity and position within the carrier flow.
Symbols & assumptions
Representative dilute-particle equation; added mass, lift, evaporation and two-way coupling may be needed.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Example 1: Particle acceleration under Stokes drag
Particle acceleration under Stokes drag
Problem & parameters. A particle starts at rest in a uniform fluid of constant speed U and experiences linear drag only.
v/U=1−e−t/τp
Solution. Solve τp v′ + v = U with v(0) = 0 using an integrating factor.
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Orange point: horizontal coordinate 2.5, calculated vertical coordinate 0.91792. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.91792 on the vertical axis. Values are rounded for display.
Scope. Dilute isolated-particle Stokes-drag reduction; no gravity or feedback on the fluid.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
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Relates conductive heat flux to temperature gradient.
Continuum / componentPhysical model
Mathematical model & short derivation
Representative formulation
q=−k∇T
Derivation / construction sketch
Assume heat flows down a temperature gradient near local thermal equilibrium.
Linearize flux in that gradient.
The proportionality coefficient is thermal conductivity k, or a tensor in anisotropic solids.
Symbols & assumptions
q is heat flux in W/m²; Fourier conduction may fail at very small scales or extremely short times.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Example 1: Steady one-dimensional diffusion benchmark
Steady one-dimensional diffusion benchmark
Problem & parameters. Solve u″ = 0 on 0 < ξ < 1 with u(0) = 1 and u(1) = 0, constant transport coefficient, and no source.
u(ξ)=1−ξ
Solution. Integrate twice to obtain u = A+Bξ. The two endpoint values give A = 1 and B = −1.
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Orange point: horizontal coordinate 0.5, calculated vertical coordinate 0.5. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 0.5 on the vertical axis. Values are rounded for display.
Scope. For numerical-method entries this is the exact target to verify against, not a computed discretization or convergence claim.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Balances thermal storage, conduction and heat sources.
Continuum / componentPhysical model
Mathematical model & short derivation
Representative formulation
ρcp∂t∂T=∇⋅(k∇T)+Q
Derivation / construction sketch
Balance energy storage in a small stationary solid volume against incoming heat and volumetric generation.
Insert Fourier’s conductive flux.
Divide by volume and take the local limit.
Symbols & assumptions
Q is heat generation per volume, cp specific heat and ρ density; moving media need advection and possibly work terms.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Use uτ = uξξ on the unit interval, zero end values, and u(ξ,0) = sin(πξ). Plot τ = 0.1.
u(ξ,τ)=sin(πξ)e−π2τ,τ=0.1
Solution. The sine satisfies both zero end values. Its second derivative is −π² times itself; the amplitude solves a′ = −π²a.
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Orange point: horizontal coordinate 0.5, calculated vertical coordinate 0.37271. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 0.37271 on the vertical axis. Values are rounded for display.
Scope. Exact PDE benchmark. For reduced bases, PINNs, and neural operators, this is a reference target, not a claimed trained or computed prediction.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
For a nonstiff ODE or semidiscrete equation when the explicit stability bound and error budget permit; usually a baseline rather than the most efficient choice.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Represents a body with one spatially uniform temperature.
Continuum / componentPhysical model
Mathematical model & short derivation
Representative formulation
mcdtdT=−hA(T−T∞)T−T∞=(T0−T∞)e−t/ττ=hAmc
Derivation / construction sketch
Assume one uniform body temperature.
Balance stored thermal energy against convective surface loss.
Integrate the first-order equation for constant properties and ambient temperature.
Symbols & assumptions
Requires small internal temperature gradients, commonly assessed by Biot number hLc/k much less than one.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Practical applications & products
Application area
Heat transfer
Practical use
Cooling of a small conductive sensor when internal gradients are negligible.
Problem & parameters. A thermal capacitance C connects through resistance R to fixed ambient temperature. Set τ = t/(RC) and y = (T−T∞)/(T0−T∞).
y(τ)=e−τ
Solution. Separate dy/dτ = −y, integrate ln(y) = −τ + C, and apply y(0) = 1.
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Orange point: horizontal coordinate 2.5, calculated vertical coordinate 0.082085. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.082085 on the vertical axis. Values are rounded for display.
Scope. One-node constant-property cooling example; multizone and multi-node networks have additional modes.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Represents heat paths and storage with connected lumped elements.
Continuum / componentPhysical model
Mathematical model & short derivation
Representative formulation
CidtdTi=Qi+j∑RijTj−Ti
Derivation / construction sketch
Partition a thermal system into nearly uniform-temperature nodes.
Assign a heat capacity C to each node and a thermal resistance R to each link.
Apply energy conservation at every node.
Symbols & assumptions
R has units K/W and C J/K. Radiation or temperature-dependent conductance makes the network nonlinear.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. A thermal capacitance C connects through resistance R to fixed ambient temperature. Set τ = t/(RC) and y = (T−T∞)/(T0−T∞).
y(τ)=e−τ
Solution. Separate dy/dτ = −y, integrate ln(y) = −τ + C, and apply y(0) = 1.
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Orange point: horizontal coordinate 2.5, calculated vertical coordinate 0.082085. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.082085 on the vertical axis. Values are rounded for display.
Scope. One-node constant-property cooling example; multizone and multi-node networks have additional modes.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Uses a heat-transfer coefficient between a surface and a fluid.
Continuum / componentPhysical model
Mathematical model & short derivation
Representative formulation
qconv=h(Ts−T∞)Qconv=hA(Ts−T∞)
Derivation / construction sketch
Represent the complicated fluid boundary layer by an effective thermal resistance.
Define h as flux divided by surface-to-bulk temperature difference.
Multiply by area for total heat flow.
Symbols & assumptions
This is a constitutive approximation; h depends on flow, geometry, fluid properties and heating conditions.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. A thermal capacitance C connects through resistance R to fixed ambient temperature. Set τ = t/(RC) and y = (T−T∞)/(T0−T∞).
y(τ)=e−τ
Solution. Separate dy/dτ = −y, integrate ln(y) = −τ + C, and apply y(0) = 1.
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Orange point: horizontal coordinate 2.5, calculated vertical coordinate 0.082085. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.082085 on the vertical axis. Values are rounded for display.
Scope. One-node constant-property cooling example; multizone and multi-node networks have additional modes.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Tracks radiation intensity through emission, absorption and scattering.
Continuum / componentPhysical model
Mathematical model & short derivation
Representative formulation
s⋅∇I=−(κa+κs)I+κaIb+4πκs∫Φ(s′→s)I(s′)dΩ′
Derivation / construction sketch
Follow radiative intensity along a ray direction s.
Subtract absorption and out-scattering.
Add thermal emission and radiation scattered into the ray from other directions.
Symbols & assumptions
Steady spectral or gray form with compatible coefficients; κa and κs are absorption and scattering coefficients and Φ is a normalized phase function.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. A steady beam traverses a homogeneous purely absorbing medium. Set τ = Σx and y = intensity / incident intensity.
y(τ)=e−τ
Solution. Separate dy/dτ = −y, integrate ln(y) = −τ + C, and apply y(0) = 1.
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Orange point: horizontal coordinate 2.5, calculated vertical coordinate 0.082085. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.082085 on the vertical axis. Values are rounded for display.
Scope. Exact absorption-only transport benchmark, without scattering or emission. For Monte Carlo transport this is the expected value, not a sampled realization.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Relates idealized surface radiant emission to the fourth power of temperature.
Continuum / componentPhysical model
Mathematical model & short derivation
Representative formulation
E=εσT4Qnet=εσA(Ts4−Tsur4)
Derivation / construction sketch
Integrate blackbody spectral emission over wavelength and outgoing directions.
The integral scales with absolute temperature to the fourth power.
Apply gray emissivity ε and subtract irradiation from a large isothermal surrounding.
Symbols & assumptions
Net formula assumes a small gray diffuse surface viewing large black surroundings; general enclosures require view factors.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. A gray surface sees a large isothermal surrounding at T*, with constant emissivity.
q/(ϵσT∗4)=θ4−1
Solution. Subtract incoming εσT*⁴ from outgoing εσT⁴. Positive net flux is outward.
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Orange point: horizontal coordinate 1.25, calculated vertical coordinate 1.4414. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 1.25, into the displayed formula to obtain 1.4414 on the vertical axis. Values are rounded for display.
Scope. Analytical solution or constitutive evaluation for the stated special case. It is not a general solution of the full model family.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Balances diffuse radiation exchange between surfaces.
Continuum / componentPhysical model
Mathematical model & short derivation
Representative formulation
Ji=εiσTi4+(1−εi)j∑FijJjQi=Ai(Ji−Gi)
Derivation / construction sketch
Define radiosity J as emitted plus reflected radiant energy.
Use view factors F to compute incident irradiation G from all surfaces.
Subtract incident from outgoing radiation to get net surface heat flow.
Symbols & assumptions
Opaque diffuse-gray surfaces, nonparticipating medium, and consistent view factors satisfying enclosure and reciprocity rules.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Two infinite parallel diffuse-gray plates have emissivities ε and 0.8 and fixed unequal temperatures.
σ(T14−T24)q=1/ϵ+1/0.8−11
Solution. Add the two surface radiation resistances and the unit view-factor space resistance; solve the radiosity balance for q.
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Orange point: horizontal coordinate 0.525, calculated vertical coordinate 0.46409. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.525, into the displayed formula to obtain 0.46409 on the vertical axis. Values are rounded for display.
Scope. Equal facing areas, view factor one, and a nonparticipating gap.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Solve heat conduction in solid and liquid regions.
Apply energy conservation to an infinitesimal layer moving with the phase boundary.
The jump in conductive flux supplies latent heat for boundary motion.
Symbols & assumptions
One-dimensional sign convention with solid on the left and liquid on the right; s is interface position and L latent heat per mass.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Take a one-phase Stefan problem whose Stefan number selects similarity constant λ = 0.5.
s/L=2λαt/L2,λ=0.5
Solution. The diffusion similarity coordinate makes the interface position s = 2λ√(αt). For this λ, the Stefan-number relation is Ste = √π λ exp(λ²) erf(λ) ≈ 0.5923.
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Orange point: horizontal coordinate 2, calculated vertical coordinate 1.4142. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 2, into the displayed formula to obtain 1.4142 on the vertical axis. Values are rounded for display.
Scope. Semi-infinite, one-phase conduction limit with a fixed boundary temperature; λ must be consistent with the material and thermal data.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
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Represents melting using enthalpy and a porous resistance in the mushy zone.
Continuum / componentPhysical model
Mathematical model & short derivation
Representative formulation
H=h+flLSmushy=−fl3+εC(1−fl)2u
Derivation / construction sketch
Include latent heat in an enthalpy H with liquid fraction fl.
Use the energy equation to update enthalpy and infer phase fraction.
Suppress velocity in partly solid cells using a porous resistance.
Symbols & assumptions
Representative enthalpy–porosity form; h is sensible enthalpy, C a mushy-zone parameter and ε a small regularization constant.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Choose the linear liquid-fraction law between solidus Ts and liquidus Tl.
fl=min[1,max(0,θ)],θ=(T−Ts)/(Tl−Ts)
Solution. Use zero fraction below Ts, linear interpolation inside the mushy interval, and unit fraction above Tl.
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Orange point: horizontal coordinate 0.5, calculated vertical coordinate 0.5. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 0.5 on the vertical axis. Values are rounded for display.
Scope. Constitutive phase-fraction example only; momentum damping and the transient enthalpy equation are not solved.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Expand elastic strain energy to quadratic order near an unstressed equilibrium.
Differentiate that energy with respect to small strain.
The resulting linear relation defines the stiffness tensor C.
Symbols & assumptions
u is displacement and σ Cauchy stress in a small-strain setting; isotropic C can be expressed using two Lamé constants.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Apply uniform uniaxial strain to a homogeneous small-strain elastic bar with traction-free lateral surfaces.
σ/E=ε
Solution. The one-dimensional constitutive law is σ = Eε; divide by E. For an orthotropic solid use its modulus along a principal material axis.
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Orange point: horizontal coordinate 0.005, calculated vertical coordinate 0.005. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.005, into the displayed formula to obtain 0.005 on the vertical axis. Values are rounded for display.
Scope. Homogeneous linear reference for truss, RVE, and FE² entries; this is not a heterogeneous microscale simulation.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Apply uniform uniaxial strain to a homogeneous small-strain elastic bar with traction-free lateral surfaces.
σ/E=ε
Solution. The one-dimensional constitutive law is σ = Eε; divide by E. For an orthotropic solid use its modulus along a principal material axis.
Open this section to load its graph, or use the full-size example link below.
Orange point: horizontal coordinate 0.005, calculated vertical coordinate 0.005. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.005, into the displayed formula to obtain 0.005 on the vertical axis. Values are rounded for display.
Scope. Homogeneous linear reference for truss, RVE, and FE² entries; this is not a heterogeneous microscale simulation.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Models large elastic deformation with a strain-energy function.
Continuum / componentPhysical model
Mathematical model & short derivation
Representative formulation
W=2μ(I1−3)−μlnJ+2λ(lnJ)2P=∂F∂W
Derivation / construction sketch
Use deformation gradient F to represent finite strain.
Choose an isotropic strain-energy function that recovers linear elasticity near F=I.
Differentiate W with respect to F to obtain first Piola stress P.
Symbols & assumptions
One compressible neo-Hookean variant; J=det F, I₁=tr(FᵀF). Other volumetric penalties are also used.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Stretch an incompressible neo-Hookean solid uniaxially with traction-free transverse faces.
σ/μ=λ2−λ−1
Solution. Incompressibility gives transverse stretches λ^−1/2. Eliminate the pressure using zero transverse stress, yielding μ(λ²−λ^−1).
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Orange point: horizontal coordinate 1.3, calculated vertical coordinate 0.92077. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 1.3, into the displayed formula to obtain 0.92077 on the vertical axis. Values are rounded for display.
Scope. Analytical solution or constitutive evaluation for the stated special case. It is not a general solution of the full model family.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Uses multiple strain invariants to fit rubber-like response.
Continuum / componentPhysical model
Mathematical model & short derivation
Representative formulation
W=C10(I1−3)+C01(I2−3)J=1
Derivation / construction sketch
Express isotropic incompressible elastic energy using invariants of FᵀF.
Retain terms linear in the first two invariants.
Differentiate the constrained energy to obtain stress plus an incompressibility pressure.
Symbols & assumptions
Two-parameter Mooney–Rivlin form; I₂ is the second invariant. Compressible versions add a volumetric energy.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Use an incompressible two-parameter Mooney–Rivlin material with C10 = C01 and μ = 2(C10+C01).
σ/μ=21(1+λ−1)(λ2−λ−1)
Solution. Differentiate the strain energy and eliminate transverse pressure. The axial stress is 2(C10+C01/λ)(λ²−1/λ).
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Orange point: horizontal coordinate 1.3, calculated vertical coordinate 0.81453. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 1.3, into the displayed formula to obtain 0.81453 on the vertical axis. Values are rounded for display.
Scope. Analytical solution or constitutive evaluation for the stated special case. It is not a general solution of the full model family.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Uses powers of principal stretches to represent nonlinear elasticity.
Continuum / componentPhysical model
Mathematical model & short derivation
Representative formulation
W=p∑αpμp(λ1αp+λ2αp+λ3αp−3)λ1λ2λ3=1
Derivation / construction sketch
Diagonalize stretch into principal stretches λᵢ.
Build an isotropic energy as a symmetric sum of stretch powers.
Fit coefficients and differentiate with respect to stretches to obtain principal stresses.
Symbols & assumptions
One common Ogden coefficient convention; other software uses different prefactors. Incompressible form shown.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Choose W = (2μ/α²)(λ1^α+λ2^α+λ3^α−3), α = 4, and incompressible uniaxial tension.
σ/μ=21(λ4−λ−2)
Solution. Set transverse stretches to λ^−1/2 and impose zero transverse stress. Then σ = (2μ/α)(λ^α−λ^−α/2).
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Orange point: horizontal coordinate 1.3, calculated vertical coordinate 1.1322. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 1.3, into the displayed formula to obtain 1.1322 on the vertical axis. Values are rounded for display.
Scope. The energy convention is stated explicitly because Ogden coefficient conventions vary.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Describes slender-beam bending while neglecting transverse shear deformation.
Continuum / componentPhysical model
Mathematical model & short derivation
Representative formulation
M=EIκEIdx4d4w=q
Derivation / construction sketch
Assume plane cross sections remain normal to the beam centerline.
Relate bending strain to curvature and integrate stress over the cross section to obtain M=EIκ.
Combine force and moment equilibrium to get the deflection equation.
Symbols & assumptions
Constant bending rigidity EI, small deflections, and a sign convention where q matches w. Shear deformation is neglected.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. A prismatic Euler–Bernoulli cantilever of length L carries a transverse tip force P.
w/(PL3/EI)=ξ2(3−ξ)/6
Solution. Use bending moment M = P(L−x). Integrate EIw″ = M and apply zero displacement and slope at the clamped end.
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Orange point: horizontal coordinate 0.5, calculated vertical coordinate 0.10417. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 0.10417 on the vertical axis. Values are rounded for display.
Scope. Analytical solution or constitutive evaluation for the stated special case. It is not a general solution of the full model family.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Includes transverse shear deformation and rotational effects.
Continuum / componentPhysical model
Mathematical model & short derivation
Representative formulation
M=EIϕ′V=κsGA(w′−ϕ)
Derivation / construction sketch
Allow cross-section rotation φ to differ from centerline slope w′.
Use their difference as transverse shear strain.
Combine bending and shear constitutive laws with beam force and moment balances.
Symbols & assumptions
κs is the shear correction factor, not curvature; consistent load signs and dynamic inertia terms complete the model.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Take an end-loaded Timoshenko cantilever with EI/(κGA L²) = 0.1.
w/(PL3/EI)=ξ2(3−ξ)/6+0.1ξ
Solution. Add the bending displacement to the shear contribution Px/(κGA). Divide by PL³/EI.
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Orange point: horizontal coordinate 0.5, calculated vertical coordinate 0.15417. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 0.15417 on the vertical axis. Values are rounded for display.
Scope. Linear prismatic beam; κ is the shear correction factor.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Describes thin-plate bending with normals remaining normal.
Continuum / componentPhysical model
Mathematical model & short derivation
Representative formulation
D∇4w=qD=12(1−ν2)Et3
Derivation / construction sketch
Assume normals to the mid-surface remain straight and normal during bending.
Integrate linear elastic bending stress through thickness t.
Use transverse equilibrium to produce the biharmonic plate equation.
Symbols & assumptions
Flat isotropic thin plate, small deflection; E is Young’s modulus and ν Poisson ratio.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Apply a single sinusoidal load mode to a simply supported rectangular plate. Plot its normalized centerline deflection.
w(x,b/2)/wmax=sin(πx/a)
Solution. A separable sin(πx/a)sin(πy/b) mode satisfies the simply supported displacement conditions. At y=b/2 it reduces to the displayed sine.
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Orange point: horizontal coordinate 0.5, calculated vertical coordinate 1. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 1 on the vertical axis. Values are rounded for display.
Scope. Kirchhoff–Love and compatible Mindlin single-mode solutions share this normalized shape but have different bending/shear amplitude formulas.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Includes transverse shear deformation in plate bending.
Continuum / componentPhysical model
Mathematical model & short derivation
Representative formulation
Q=κsGt(∇w−θ)M=Dbκ(θ)
Derivation / construction sketch
Allow plate rotations θ to differ from the gradient of transverse displacement.
Use the difference to generate shear resultants Q.
Combine shear and bending resultants with equilibrium.
Symbols & assumptions
Db is the plate bending stiffness matrix and κ(θ) the curvature vector; sign conventions and shear correction must be consistent.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Apply a single sinusoidal load mode to a simply supported rectangular plate. Plot its normalized centerline deflection.
w(x,b/2)/wmax=sin(πx/a)
Solution. A separable sin(πx/a)sin(πy/b) mode satisfies the simply supported displacement conditions. At y=b/2 it reduces to the displayed sine.
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Orange point: horizontal coordinate 0.5, calculated vertical coordinate 1. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 1 on the vertical axis. Values are rounded for display.
Scope. Kirchhoff–Love and compatible Mindlin single-mode solutions share this normalized shape but have different bending/shear amplitude formulas.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
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Combines membrane and bending behavior on a curved surface.
Continuum / componentPhysical model
Mathematical model & short derivation
Representative formulation
N=Aε0+BκM=Bε0+Dκ
Derivation / construction sketch
Describe in-plane strain as a mid-surface strain plus a thickness-dependent curvature term.
Integrate stresses through thickness to obtain membrane forces N and moments M.
The integrals define extensional, coupling and bending stiffnesses A, B and D.
Symbols & assumptions
Representative linear shell/laminate constitutive form; geometry supplies membrane and curvature relations and equilibrium.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. A thin spherical shell of radius R and thickness t carries uniform internal pressure p.
σ/(E)=21[pR/(Et)]
Solution. Balance pressure on a hemisphere against the circumferential membrane force: pπR² = 2πRtσ.
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Orange point: horizontal coordinate 0.01, calculated vertical coordinate 0.005. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.01, into the displayed formula to obtain 0.005 on the vertical axis. Values are rounded for display.
Scope. Thin-shell membrane approximation, away from supports and local bending disturbances.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
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Represents a structure with axial-force members joined at idealized nodes.
Continuum / componentPhysical model
Mathematical model & short derivation
Representative formulation
N=EALΔLk=LEA
Derivation / construction sketch
Assume a straight member carries only axial force.
Use axial strain ΔL/L and linear elasticity σ=Eε.
Multiply by cross-sectional area to obtain force and axial stiffness.
Symbols & assumptions
Ideal pin-jointed truss member with small strain; bending and joint stiffness are omitted.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Apply uniform uniaxial strain to a homogeneous small-strain elastic bar with traction-free lateral surfaces.
σ/E=ε
Solution. The one-dimensional constitutive law is σ = Eε; divide by E. For an orthotropic solid use its modulus along a principal material axis.
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Orange point: horizontal coordinate 0.005, calculated vertical coordinate 0.005. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.005, into the displayed formula to obtain 0.005 on the vertical axis. Values are rounded for display.
Scope. Homogeneous linear reference for truss, RVE, and FE² entries; this is not a heterogeneous microscale simulation.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
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Represent slender or thin structures dominated by tension.
Continuum / componentPhysical model
Mathematical model & short derivation
Representative formulation
dsd(Tt^)+f=0
Derivation / construction sketch
Consider a small segment of a flexible cable.
Its internal force acts along the local tangent t̂ because bending resistance is neglected.
Balance the change in tension vector with distributed external load f.
Symbols & assumptions
Cable equilibrium with arc length s and tension T. Membranes use the analogous surface-divergence balance of in-plane stress resultants.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. An ideal flexible cable supports its own uniform weight per arc length; choose a = horizontal tension / weight per length.
y/a=cosh(x/a)−1
Solution. Force balance gives y″ = √(1+y′²)/a. Symmetry at the lowest point integrates to the catenary.
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Orange point: horizontal coordinate 0, calculated vertical coordinate 0. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 0 on the vertical axis. Values are rounded for display.
Scope. Self-weight catenary, not the parabolic approximation for uniform load per horizontal span.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
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Uses deviatoric stress to define yielding in an isotropic ductile material.
Continuum / componentPhysical model
Mathematical model & short derivation
Representative formulation
f=3s:s/2−σy=0ε˙p=λ˙∂σ∂f
Derivation / construction sketch
Remove hydrostatic stress to obtain deviatoric stress s.
Use its second invariant to define an isotropic yield surface.
Combine the yield condition with a flow rule, hardening law and consistency condition.
Symbols & assumptions
Small-strain associative J2 plasticity shown; λ̇ is a plastic multiplier and σy may evolve with plastic strain.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Example 1: Ideal uniaxial elastic-perfectly-plastic response
Ideal uniaxial elastic-perfectly-plastic response
Problem & parameters. Load monotonically in uniaxial tension from an unstressed state, with no hardening.
σ/σy=min(Eε/σy,1)
Solution. Use Hooke’s law until σ = σy. Further strain is plastic while stress stays at σy.
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Orange point: horizontal coordinate 1.5, calculated vertical coordinate 1. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 1.5, into the displayed formula to obtain 1 on the vertical axis. Values are rounded for display.
Scope. Uniaxial case where J2 and Tresca coincide; multiaxial yield surfaces differ.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
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Compute maximum shear stress as half the largest principal-stress difference.
Set that shear stress equal to the shear stress at uniaxial yield.
The resulting principal-stress surface is the Tresca criterion.
Symbols & assumptions
σᵢ are principal stresses; a flow rule and hardening relation are needed for post-yield deformation.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Example 1: Ideal uniaxial elastic-perfectly-plastic response
Ideal uniaxial elastic-perfectly-plastic response
Problem & parameters. Load monotonically in uniaxial tension from an unstressed state, with no hardening.
σ/σy=min(Eε/σy,1)
Solution. Use Hooke’s law until σ = σy. Further strain is plastic while stress stays at σy.
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Orange point: horizontal coordinate 1.5, calculated vertical coordinate 1. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 1.5, into the displayed formula to obtain 1 on the vertical axis. Values are rounded for display.
Scope. Uniaxial case where J2 and Tresca coincide; multiaxial yield surfaces differ.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
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Represent pressure sensitivity through the first stress invariant I₁.
Represent shear loading through the second deviatoric invariant J₂.
Choose α and k to fit the desired frictional yield envelope.
Symbols & assumptions
Tension-positive convention shown; parameter signs depend on convention and calibration. Flow can be nonassociated.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Example 1: Pressure-dependent Drucker–Prager strength
Pressure-dependent Drucker–Prager strength
Problem & parameters. Define the illustrative yield line q−0.5p−c=0 with compression-positive pressure p.
q/c=1+0.5(p/c)
Solution. Solve the stated yield function for q.
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Orange point: horizontal coordinate 2, calculated vertical coordinate 2. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 2, into the displayed formula to obtain 2 on the vertical axis. Values are rounded for display.
Scope. A specified pressure/deviatoric convention and slope; different parameter mappings to friction angle exist.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
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Relates frictional shear strength to normal stress and cohesion.
Continuum / componentPhysical model
Mathematical model & short derivation
Representative formulation
τf=c+σntanϕ
Derivation / construction sketch
Resolve normal and shear tractions on a possible failure plane.
Assume frictional resistance grows linearly with compressive normal stress.
Add cohesion c to obtain the failure envelope.
Symbols & assumptions
σn is compression-positive, φ friction angle. Tensile cutoff, dilation and plastic flow require additional assumptions.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Use cohesion c > 0 and friction angle 30 degrees.
τf/c=1+(σn/c)tan(30∘)
Solution. Insert the normal stress into τf = c+σn tan φ with compression positive.
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Orange point: horizontal coordinate 2, calculated vertical coordinate 2.1547. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 2, into the displayed formula to obtain 2.1547 on the vertical axis. Values are rounded for display.
Scope. Analytical solution or constitutive evaluation for the stated special case. It is not a general solution of the full model family.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
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Uses empirical strain, strain-rate and temperature factors.
Continuum / componentPhysical model
Mathematical model & short derivation
Representative formulation
σy=(A+Bεpn)[1+Cln(ε˙p/ε˙0)][1−(T∗)m]
Derivation / construction sketch
Separate empirical effects of strain hardening, strain-rate sensitivity and thermal softening.
Fit each factor to suitable tests.
Multiply the factors to obtain flow stress over the calibrated regime.
Symbols & assumptions
T*=(T−Tref)/(Tmelt−Tref), usually bounded to an intended range. This is an empirical constitutive law, not a first-principles derivation.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Set B/A = 0.5, n = 0.5, strain rate equal to its reference value, and homologous temperature zero.
σ/A=1+0.5εp
Solution. The rate and thermal factors become one. Evaluate the remaining A+Bεpⁿ hardening term.
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Orange point: horizontal coordinate 0.5, calculated vertical coordinate 1.3536. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 1.3536 on the vertical axis. Values are rounded for display.
Scope. Illustrative constants, not a calibrated metal response.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
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Represents plastic flow through crystallographic slip systems.
Continuum / componentPhysical model
Mathematical model & short derivation
Representative formulation
Lp=α∑γ˙αsα⊗mατα=σ:(sα⊗mα)
Derivation / construction sketch
Decompose deformation into elastic lattice distortion and crystallographic slip.
Resolve stress onto each slip direction sα and plane normal mα.
Use slip-rate and hardening laws to assemble the plastic velocity gradient Lp.
Symbols & assumptions
Representative small-elastic-strain slip-system form; finite-strain stress measures and lattice rotation require a consistent formulation.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. For positive resolved shear choose rate sensitivity m = 0.2 and fixed slip resistance g.
γ˙/γ˙0=(τ/g)5
Solution. Evaluate γ̇ = γ̇0(τ/g)^(1/m).
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Orange point: horizontal coordinate 0.75, calculated vertical coordinate 0.2373. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.75, into the displayed formula to obtain 0.2373 on the vertical axis. Values are rounded for display.
Scope. Single-system constitutive evaluation; lattice rotation and hardening are held fixed.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
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Combines an elastic spring and viscous dashpot in series.
Continuum / componentPhysical model
Mathematical model & short derivation
Representative formulation
ε˙=Eσ˙+ησ
Derivation / construction sketch
Put a spring and dashpot in series so they share the same stress.
Add their strains.
Differentiate and use the elastic and viscous constitutive laws.
Symbols & assumptions
For fixed total strain, stress decays with relaxation time η/E. E is spring modulus and η dashpot viscosity.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Apply a step strain ε₀ to a Maxwell spring–dashpot series element and hold it fixed. Scale stress by Eε₀ and time by η/E.
y(τ)=e−τ
Solution. Separate dy/dτ = −y, integrate ln(y) = −τ + C, and apply y(0) = 1.
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Orange point: horizontal coordinate 2.5, calculated vertical coordinate 0.082085. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.082085 on the vertical axis. Values are rounded for display.
Scope. Exact one-mode reduction with constant coefficients; additional coupled physics is excluded.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
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Combines an elastic spring and viscous dashpot in parallel.
Continuum / componentPhysical model
Mathematical model & short derivation
Representative formulation
σ=Eε+ηε˙
Derivation / construction sketch
Put a spring and dashpot in parallel so they share strain.
Add their stresses.
Substitute Hooke and Newton constitutive laws.
Symbols & assumptions
A constant stress produces delayed creep toward σ/E with time constant η/E; ideal instantaneous strain jumps require infinite dashpot stress.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Apply constant stress σ₀ at t = 0 to an initially undeformed parallel spring and dashpot.
Eε/σ0=1−e−Et/η
Solution. Solve ηε′+Eε = σ₀ with zero initial strain.
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Orange point: horizontal coordinate 2.5, calculated vertical coordinate 0.91792. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.91792 on the vertical axis. Values are rounded for display.
Scope. Analytical solution or constitutive evaluation for the stated special case. It is not a general solution of the full model family.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
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Place an equilibrium spring E₀ in parallel with a Maxwell branch E₁,η.
Write total stress as E₀ε plus the branch stress.
Eliminate branch stress using the Maxwell constitutive equation.
Symbols & assumptions
This standard-linear-solid arrangement has instantaneous modulus E₀+E₁ and long-time modulus E₀.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Practical applications & products
Application area
Plasticity, damage & durability
Practical use
Creep and relaxation over one characteristic time scale.
Problem & parameters. Apply a fixed strain step to a standard linear solid with relaxed modulus E∞ = 0.4E0.
σ/(E0ε0)=0.4+0.6e−t/τ
Solution. Its relaxation modulus is E∞+(E0−E∞)exp(−t/τ). Multiply by the imposed strain.
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Orange point: horizontal coordinate 2.5, calculated vertical coordinate 0.44925. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.44925 on the vertical axis. Values are rounded for display.
Scope. Analytical solution or constitutive evaluation for the stated special case. It is not a general solution of the full model family.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
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Represent thermally activated creep with an Arrhenius temperature factor.
Fit a power-law dependence on stress.
Combine the two to describe a calibrated steady creep regime.
Symbols & assumptions
Uniaxial Norton-type law; A,n,Q depend on material and mechanism. Primary and tertiary creep need other terms.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. At fixed temperature use Norton exponent n = 3 and reference rate Aσ*³.
ε˙/ε˙∗=(σ/σ∗)3
Solution. Substitute the stress into ε̇ = Aσ³.
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Orange point: horizontal coordinate 1, calculated vertical coordinate 1. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 1, into the displayed formula to obtain 1 on the vertical axis. Values are rounded for display.
Scope. Steady creep constitutive law at fixed material state and temperature.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
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Uses crack-tip intensity parameters in an elastic body.
Continuum / componentPhysical model
Mathematical model & short derivation
Representative formulation
σij≈2πrKIfij(θ)G=E′KI2
Derivation / construction sketch
Solve elasticity near a crack tip and retain the leading singular field.
Its amplitude is the mode-I stress intensity KI.
Relate the field energy release to KI using elastic energy balance.
Symbols & assumptions
E′=E for plane stress and E/(1−ν²) for plane strain. Small-scale yielding and an appropriate crack geometry are required.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Example 1: Crack-tip opening stress on the forward ray
Crack-tip opening stress on the forward ray
Problem & parameters. Use the leading mode-I elastic crack-tip field on θ = 0.
σyy/(KI/2πℓ)=(r/ℓ)−1/2
Solution. The angular factor equals one on the forward ray. Evaluate KI/√(2πr).
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Orange point: horizontal coordinate 1.025, calculated vertical coordinate 0.98773. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 1.025, into the displayed formula to obtain 0.98773 on the vertical axis. Values are rounded for display.
Scope. Near-tip linear-elastic asymptotic field, outside the process zone; the singular tip itself is excluded.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
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Uses traction-separation relations across a fracture process zone.
Continuum / componentPhysical model
Mathematical model & short derivation
Representative formulation
t=t(δ)Gc=∫0δft(δ)dδ
Derivation / construction sketch
Replace a singular crack-tip region with a finite traction-separation law.
Allow traction t to rise and then soften as separation δ grows.
The area under the curve gives the energy required to create unit crack area.
Symbols & assumptions
Representative single-mode cohesive law; mixed-mode interaction, unloading and irreversibility need specifications.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Choose peak traction at half the complete-separation opening and linear loading/softening branches.
t/tmax={2d2(1−d)d≤0.5d>0.5
Solution. Connect (0,0), (δc/2,tmax), and (δc,0). The work of separation is the triangle area tmaxδc/2.
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Orange point: horizontal coordinate 0.5, calculated vertical coordinate 1. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 1 on the vertical axis. Values are rounded for display.
Scope. Monotonic prescribed cohesive law; unloading and mixed-mode effects are excluded.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
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Represents cracks with a continuous damage-like field.
Continuum / componentPhysical model
Mathematical model & short derivation
Representative formulation
Π=∫[g(d)ψe(ε)+Gc(d2/(2ℓ)+ℓ∣∇d∣2/2)]dV−Wext
Derivation / construction sketch
Approximate a sharp crack surface energy with a diffuse damage field d.
Degrade elastic energy using g(d).
Vary the total energy with respect to displacement and damage, imposing irreversibility.
Symbols & assumptions
Representative AT2 phase-field fracture energy; ℓ controls regularization width. Tension-compression splitting and history treatment affect results.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Minimize the isolated AT2 crack-surface functional with d(0)=1 and d→0 far from the crack, without mechanical driving away from x=0.
d(x)=e−∣x∣/ℓ
Solution. The Euler equation is d−ℓ²d″=0 on each half-line. Select decaying exponentials and enforce symmetry.
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Orange point: horizontal coordinate 0, calculated vertical coordinate 1. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 1 on the vertical axis. Values are rounded for display.
Scope. Stationary isolated crack-profile benchmark, not a coupled fracture-growth solution.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
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Relates cyclic crack-growth rate to stress-intensity-factor range.
Continuum / componentPhysical model
Mathematical model & short derivation
Representative formulation
dNda=C(ΔK)m
Derivation / construction sketch
Measure crack extension per loading cycle in the stable growth region.
Plot growth rate against stress-intensity range on logarithmic axes.
Fit the approximately linear region to obtain C and m.
Symbols & assumptions
Empirical Paris law; it excludes near-threshold and near-instability behavior unless extended. Load ratio and environment matter.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Example 1: Paris-law crack growth for exponent two
Paris-law crack growth for exponent two
Problem & parameters. Use da/dN=C(ΔK)² and ΔK=Δσ√(πa) with constant stress range and geometry factor one.
a/a0=eN/N∗,N∗=(CΔσ2π)−1
Solution. Substitute ΔK to obtain da/dN = CΔσ²πa. Separate variables and apply a(0)=a₀.
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Orange point: horizontal coordinate 0.75, calculated vertical coordinate 2.117. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.75, into the displayed formula to obtain 2.117 on the vertical axis. Values are rounded for display.
Scope. Only within the Paris regime; threshold, instability, and changing geometry are excluded.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
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Adds fractions of fatigue life consumed by load cycles.
Continuum / componentPhysical model
Mathematical model & short derivation
Representative formulation
D=i∑Nininominal failure at D≈1
Derivation / construction sketch
Estimate constant-amplitude life Nᵢ for each load level.
Treat nᵢ cycles at that level as consuming fraction nᵢ/Nᵢ of life.
Add fractions across the load history.
Symbols & assumptions
Linear Miner accumulation ignores sequence and interaction effects; D=1 is an engineering approximation, not a universal threshold.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Practical applications & products
Application area
Plasticity, damage & durability
Practical use
Approximate damage under variable-amplitude loading.
Problem & parameters. Apply constant-amplitude cycles with a fixed fatigue life Nf.
D=n/Nf
Solution. Miner’s sum has one term n/Nf; the conventional failure threshold is D=1.
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Orange point: horizontal coordinate 0.5, calculated vertical coordinate 0.5. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 0.5 on the vertical axis. Values are rounded for display.
Scope. Linear accumulation hypothesis, not a physical guarantee of failure at exactly D=1.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Relates wear volume to load, sliding distance and hardness.
Continuum / componentPhysical model
Mathematical model & short derivation
Representative formulation
Vwear=HKWs
Derivation / construction sketch
Assume material loss scales with normal load W and sliding distance s.
Normalize by hardness H to reflect resistance to plastic contact deformation.
Introduce empirical dimensionless wear coefficient K.
Symbols & assumptions
Archard mild-wear form; mechanisms, lubrication and changing contact conditions can invalidate a constant K.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Hold wear coefficient k, normal force W, and hardness H constant.
VH/(kWs∗)=s/s∗
Solution. Integrate dV/ds = kW/H from zero initial wear.
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Orange point: horizontal coordinate 2.5, calculated vertical coordinate 2.5. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 2.5 on the vertical axis. Values are rounded for display.
Scope. Steady Archard wear regime with no changes in contact, debris, or material properties.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Balances forces and moments on translating and rotating bodies.
Component / systemPhysical model
Mathematical model & short derivation
Representative formulation
ma=∑FIω˙+ω×(Iω)=∑M
Derivation / construction sketch
Apply linear momentum balance to the center of mass.
Apply angular momentum balance about that center.
Express angular momentum in body coordinates, introducing the rotating-frame cross product.
Symbols & assumptions
Rigid-body inertia tensor I is constant in body coordinates; forces and moments must be expressed consistently.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. A rigid body starts at rest with constant net force-to-mass ratio 1 m/s² along one axis and zero net torque.
x(t)=21at2,a=1m/s2
Solution. Newton’s law gives constant acceleration. Integrate twice with zero initial position and velocity.
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Orange point: horizontal coordinate 2.5, calculated vertical coordinate 3.125. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 3.125 on the vertical axis. Values are rounded for display.
Scope. Single translational degree of freedom; the remaining forces, torques, and rotational motion are set to zero.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Derives motion from kinetic and potential energy with constraints.
Component / systemPhysical model
Mathematical model & short derivation
Representative formulation
dtd(∂q˙i∂L)−∂qi∂L=QiL=T−V
Derivation / construction sketch
Write action as the time integral of kinetic minus potential energy.
Vary the path while holding its endpoints fixed.
Integrate by parts and include nonconservative generalized forces Qᵢ.
Symbols & assumptions
qᵢ are generalized coordinates; constraints must be eliminated or enforced with multipliers.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Choose a single unconstrained linear mode with zero damping, initial displacement A, and zero velocity.
q/A=cos(ωt)
Solution. Either force balance or the quadratic energy gives q″+ω²q=0. Apply the initial conditions to select the cosine.
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Orange point: horizontal coordinate 6.2832, calculated vertical coordinate 1. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 6.2832, into the displayed formula to obtain 1 on the vertical axis. Values are rounded for display.
Scope. Exact single harmonic mode; multibody constraints and other modal couplings are absent.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Describes dynamics in generalized coordinates and momenta.
Component / systemPhysical model
Mathematical model & short derivation
Representative formulation
q˙i=∂pi∂Hp˙i=−∂qi∂H
Derivation / construction sketch
Define momenta pᵢ = ∂L/∂q̇ᵢ.
Perform the Legendre transform H=Σpᵢq̇ᵢ−L.
Rearrange the Euler–Lagrange equations into first-order phase-space equations.
Symbols & assumptions
The usual transform assumes a nondegenerate velocity Hessian; constrained systems require extra treatment.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Choose a single unconstrained linear mode with zero damping, initial displacement A, and zero velocity.
q/A=cos(ωt)
Solution. Either force balance or the quadratic energy gives q″+ω²q=0. Apply the initial conditions to select the cosine.
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Orange point: horizontal coordinate 6.2832, calculated vertical coordinate 1. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 6.2832, into the displayed formula to obtain 1 on the vertical axis. Values are rounded for display.
Scope. Exact single harmonic mode; multibody constraints and other modal couplings are absent.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Represents inertia, stiffness and dissipation with lumped elements.
Component / systemPhysical model
Mathematical model & short derivation
Representative formulation
mx¨+cx˙+kx=F(t)
Derivation / construction sketch
Identify inertial, viscous and elastic forces on a lumped mass.
Use −cẋ and −kx as resisting forces.
Apply Newton’s second law and collect terms.
Symbols & assumptions
Linear single-degree-of-freedom model; c is viscous damping, k stiffness and F external force.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Choose a single unconstrained linear mode with zero damping, initial displacement A, and zero velocity.
q/A=cos(ωt)
Solution. Either force balance or the quadratic energy gives q″+ω²q=0. Apply the initial conditions to select the cosine.
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Orange point: horizontal coordinate 6.2832, calculated vertical coordinate 1. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 6.2832, into the displayed formula to obtain 1 on the vertical axis. Values are rounded for display.
Scope. Exact single harmonic mode; multibody constraints and other modal couplings are absent.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Expands linear structural response into vibration modes.
Component / systemPhysical model
Mathematical model & short derivation
Representative formulation
u=ΦqMrq¨+Crq˙+Krq=ΦTf
Derivation / construction sketch
Solve the generalized eigenproblem Kφ=ω²Mφ.
Expand displacement in selected eigenvectors Φ.
Project the full equations onto their span; proportional damping gives decoupled modal equations.
Symbols & assumptions
Linear system with consistent mass and stiffness; truncated modes omit part of the dynamic response.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Choose a single unconstrained linear mode with zero damping, initial displacement A, and zero velocity.
q/A=cos(ωt)
Solution. Either force balance or the quadratic energy gives q″+ω²q=0. Apply the initial conditions to select the cosine.
Open this section to load its graph, or use the full-size example link below.
Orange point: horizontal coordinate 6.2832, calculated vertical coordinate 1. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 6.2832, into the displayed formula to obtain 1 on the vertical axis. Values are rounded for display.
Scope. Exact single harmonic mode; multibody constraints and other modal couplings are absent.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Expand a symmetric restoring force around equilibrium.
Keep its linear and leading cubic terms.
Add inertia, damping and periodic forcing.
Symbols & assumptions
Duffing model; α controls hardening or softening. A negative cubic term alone does not define a globally stable potential.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. For positive linear and cubic stiffness choose ℓ=√(k/β). Find the force needed to hold a static displacement.
F/(kℓ)=q+q3,q=x/ℓ
Solution. Set velocity and acceleration to zero in the Duffing equation. Normalize F=kx+βx³.
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Orange point: horizontal coordinate 0, calculated vertical coordinate 0. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 0 on the vertical axis. Values are rounded for display.
Scope. Static hardening equilibrium curve, not a forced nonlinear transient or resonance calculation.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Couples rigid or flexible bodies through joints and force elements.
Component / systemPhysical model
Mathematical model & short derivation
Representative formulation
M(q)q¨+h(q,q˙)=Q+J(q)TλΦ(q)=0
Derivation / construction sketch
Write the kinetic energy of all bodies in generalized coordinates.
Apply Lagrange’s equations.
Enforce joint constraints Φ with multipliers λ and constraint Jacobian J.
Symbols & assumptions
Rigid or flexible bodies need appropriate coordinates and constitutive forces; numerical constraint drift must be managed.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Choose a single unconstrained linear mode with zero damping, initial displacement A, and zero velocity.
q/A=cos(ωt)
Solution. Either force balance or the quadratic energy gives q″+ω²q=0. Apply the initial conditions to select the cosine.
Open this section to load its graph, or use the full-size example link below.
Orange point: horizontal coordinate 6.2832, calculated vertical coordinate 1. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 6.2832, into the displayed formula to obtain 1 on the vertical axis. Values are rounded for display.
Scope. Exact single harmonic mode; multibody constraints and other modal couplings are absent.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Describes small pressure perturbations about an equilibrium state.
Component / systemPhysical model
Mathematical model & short derivation
Representative formulation
∂t2∂2p′=c2∇2p′
Derivation / construction sketch
Linearize continuity and momentum about a stationary uniform fluid.
Close small density perturbations with p′=c²ρ′.
Differentiate continuity in time and eliminate velocity divergence.
Symbols & assumptions
Small-amplitude lossless acoustics in a uniform medium; background flow and dissipation add terms.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Use a one-dimensional sinusoidal wave in a uniform, lossless linear medium. Plot the normalized field at time zero.
u(ξ,0)=sin(2πξ)
Solution. A sinusoidal traveling-wave solution is u = sin[2π(ξ−τ)]. Set τ = 0 to obtain the plotted snapshot.
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Orange point: horizontal coordinate 0.5, calculated vertical coordinate 1.2246e-16. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 1.2246e-16 on the vertical axis. Values are rounded for display.
Scope. An acoustic, electromagnetic, elastic, or linear Alfvén-wave reference as appropriate. For MHD this is the small transverse perturbation of a uniform magnetized equilibrium; for FDTD it is an exact target, not a discretized result.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Represents harmonic acoustic fields at one frequency.
Component / systemPhysical model
Mathematical model & short derivation
Representative formulation
∇2P+k2P=0k=cω
Derivation / construction sketch
Assume a harmonic pressure p′=Re[P(x)e^(−iωt)].
Substitute it into the acoustic wave equation.
Cancel the common time factor to obtain a spatial Helmholtz equation.
Symbols & assumptions
Frequency-domain homogeneous-medium form; impedance and radiation boundary conditions determine the solution.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Solve p″+k²p=0 with pressure-release endpoints and choose the first nonzero eigenmode.
p(x)/P=sin(πx/L),k=π/L
Solution. Both endpoint conditions select kL=π. Normalize the remaining arbitrary amplitude by its maximum.
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Orange point: horizontal coordinate 0.5, calculated vertical coordinate 1. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 1 on the vertical axis. Values are rounded for display.
Scope. Analytical solution or constitutive evaluation for the stated special case. It is not a general solution of the full model family.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Use a one-dimensional sinusoidal wave in a uniform, lossless linear medium. Plot the normalized field at time zero.
u(ξ,0)=sin(2πξ)
Solution. A sinusoidal traveling-wave solution is u = sin[2π(ξ−τ)]. Set τ = 0 to obtain the plotted snapshot.
Open this section to load its graph, or use the full-size example link below.
Orange point: horizontal coordinate 0.5, calculated vertical coordinate 1.2246e-16. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 1.2246e-16 on the vertical axis. Values are rounded for display.
Scope. An acoustic, electromagnetic, elastic, or linear Alfvén-wave reference as appropriate. For MHD this is the small transverse perturbation of a uniform magnetized equilibrium; for FDTD it is an exact target, not a discretized result.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Couples electric and magnetic fields with charges and currents.
Cross-scalePhysical model
Mathematical model & short derivation
Representative formulation
∇⋅D=ρf∇⋅B=0∇×E=−∂tB∇×H=Jf+∂tD
Derivation / construction sketch
Express electric and magnetic flux laws in differential form.
Combine Faraday induction with Ampère’s law including displacement current.
Close the system with material relations such as D=εE and B=μH.
Symbols & assumptions
ρf and Jf are free charge and current; these fundamental laws require material and boundary data rather than a derivation from a simpler classical model.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Use a one-dimensional sinusoidal wave in a uniform, lossless linear medium. Plot the normalized field at time zero.
u(ξ,0)=sin(2πξ)
Solution. A sinusoidal traveling-wave solution is u = sin[2π(ξ−τ)]. Set τ = 0 to obtain the plotted snapshot.
Open this section to load its graph, or use the full-size example link below.
Orange point: horizontal coordinate 0.5, calculated vertical coordinate 1.2246e-16. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 1.2246e-16 on the vertical axis. Values are rounded for display.
Scope. An acoustic, electromagnetic, elastic, or linear Alfvén-wave reference as appropriate. For MHD this is the small transverse perturbation of a uniform magnetized equilibrium; for FDTD it is an exact target, not a discretized result.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Assume no time-varying magnetic induction so ∇×E=0.
Introduce scalar electric potential φ.
Substitute D=εE into Gauss’s law.
Symbols & assumptions
Spatially varying or anisotropic ε can remain inside the divergence; nonlinear dielectric response requires a corresponding constitutive law.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Example 1: Uniform-charge potential between grounded planes
Uniform-charge potential between grounded planes
Problem & parameters. Solve φ″ = −ρ/ε for constant charge density between φ(0)=φ(L)=0.
ϕ/(ρL2/ϵ)=21ξ(1−ξ)
Solution. Integrate the constant second derivative twice. The grounded endpoints fix both integration constants.
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Orange point: horizontal coordinate 0.5, calculated vertical coordinate 0.125. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 0.125 on the vertical axis. Values are rounded for display.
Scope. Analytical solution or constitutive evaluation for the stated special case. It is not a general solution of the full model family.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Represents steady magnetic fields driven by currents and magnetization.
Cross-scalePhysical model
Mathematical model & short derivation
Representative formulation
∇×H=J∇⋅B=0B=μH
Derivation / construction sketch
Set time derivatives to zero in Maxwell’s equations.
Preserve current-driven magnetic circulation and absence of magnetic monopoles.
Combine with a magnetic material law and, if useful, a vector potential B=∇×A.
Symbols & assumptions
Steady-current model; saturation and hysteresis require nonlinear or path-dependent material relations.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Consider the exterior of a long straight wire of radius a carrying steady current I in vacuum.
B/(μ0I/2πa)=a/r
Solution. Ampère’s law around a circle gives 2πrB=μ0I.
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Orange point: horizontal coordinate 3, calculated vertical coordinate 0.33333. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 3, into the displayed formula to obtain 0.33333 on the vertical axis. Values are rounded for display.
Scope. Exterior field of an ideal long wire; end effects are excluded.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Models induced conducting currents in a time-varying magnetic field.
Cross-scalePhysical model
Mathematical model & short derivation
Representative formulation
∇×(μ−1∇×A)+σ(∂tA+∇ϕ)=Js
Derivation / construction sketch
Write B=∇×A and E=−∂tA−∇φ.
Use Ohm’s law Jeddy=σE.
Substitute into Ampère’s law while neglecting displacement current.
Symbols & assumptions
Quasistatic conducting-medium form; gauge conditions and charge conservation are needed to determine A and φ.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. A sinusoidal magnetic field penetrates a homogeneous conducting half-space with skin depth δ.
∣B(x)∣/∣B(0)∣=e−x/δ
Solution. The diffusion equation at angular frequency ω has a decaying complex solution exp[−(1+i)x/δ]. Its amplitude is exp(−x/δ).
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Orange point: horizontal coordinate 2.5, calculated vertical coordinate 0.082085. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.082085 on the vertical axis. Values are rounded for display.
Scope. Linear conductor with constant conductivity and permeability; displacement current neglected.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Uses reluctance and magnetomotive force in lumped magnetic paths.
Cross-scalePhysical model
Mathematical model & short derivation
Representative formulation
Φ=RNIR=μAℓ
Derivation / construction sketch
Integrate Ampère’s law around a magnetic path.
Assume approximately uniform flux through cross-sectional area A.
Combine B=μH with Φ=BA to obtain the reluctance relation.
Symbols & assumptions
NI is magnetomotive force; leakage, fringing, saturation and multiple flux paths require corrections or a network.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Use a single magnetic circuit of fixed reluctance ℛ with no leakage.
Φ/Φ∗=(NI)/(RΦ∗)
Solution. Solve NI=ℛΦ for flux.
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Orange point: horizontal coordinate 1.5, calculated vertical coordinate 1.5. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 1.5, into the displayed formula to obtain 1.5 on the vertical axis. Values are rounded for display.
Scope. Linear unsaturated material and fixed geometry.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Represents path-dependent magnetization with phenomenological parameters.
Cross-scalePhysical model
Mathematical model & short derivation
Representative formulation
Man=Ms[coth(He/a)−a/He]He=H+αM
Derivation / construction sketch
Use an anhysteretic magnetization curve as the reversible equilibrium target.
Introduce effective-field coupling and a pinning-controlled irreversible component.
Combine reversible and irreversible magnetization to generate history-dependent loops.
Symbols & assumptions
This is the anhysteretic backbone of Jiles–Atherton, not the complete hysteresis law; pinning k, reversibility c and branch rules are also required.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Evaluate the Langevin-form anhysteretic component of a Jiles–Atherton model, using its zero-field limit M=0.
Man/Ms=cothh−1/h
Solution. Insert h=He/a into Ms[coth(h)−1/h]. The apparent singularity is removable; the small-field slope is 1/3.
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Orange point: horizontal coordinate 0, calculated vertical coordinate 0. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 0 on the vertical axis. Values are rounded for display.
Scope. Anhysteretic reference only, not the history-dependent hysteresis loop.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Insert a rapidly oscillating wave ansatz into the wave equation.
Keep the leading short-wavelength terms to obtain the eikonal equation.
Rays follow normals to phase surfaces and obey the ray equation.
Symbols & assumptions
S is optical phase path, n refractive index and s arc length; diffraction is neglected when wavelength is small relative to geometry.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. A ray crosses a plane interface from index 1 into index 1.5.
θ2=arcsin[sin(θ1)/1.5]
Solution. Use Snell’s law n1 sin θ1=n2 sin θ2 and solve for the refracted angle.
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Orange point: horizontal coordinate 40, calculated vertical coordinate 25.374. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 40, into the displayed formula to obtain 25.374 on the vertical axis. Values are rounded for display.
Scope. Analytical solution or constitutive evaluation for the stated special case. It is not a general solution of the full model family.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
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Uses a scalar wave approximation for light diffraction.
Cross-scalePhysical model
Mathematical model & short derivation
Representative formulation
U(P)≈iλ1∫apertureU(Q)reikrK(θ)dA
Derivation / construction sketch
Represent a scalar wave as contributions from an aperture boundary.
Apply a Green-function surface integral and suitable aperture approximations.
Sum secondary-wave contributions with phase delay and an obliquity factor K.
Symbols & assumptions
Representative diffraction integral; assumptions differ among Kirchhoff, Fresnel and Fraunhofer forms. Polarization is neglected.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Illuminate a slit of width a uniformly with monochromatic coherent light and observe the Fraunhofer pattern.
I/I0=[usinu]2
Solution. Integrate the phase factor across the slit to obtain sinc amplitude; square its magnitude. Use the continuous limit I/I0=1 at u=0.
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Orange point: horizontal coordinate 0, calculated vertical coordinate 1. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 1 on the vertical axis. Values are rounded for display.
Scope. Analytical solution or constitutive evaluation for the stated special case. It is not a general solution of the full model family.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
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Represents a paraxial beam with a Gaussian transverse profile.
Cross-scalePhysical model
Mathematical model & short derivation
Representative formulation
w(z)=w01+(z/zR)2zR=λπw02
Derivation / construction sketch
Factor a rapidly varying axial phase from a scalar wave.
Neglect the second axial derivative of the slowly varying envelope to get the paraxial equation.
A Gaussian ansatz gives the beam-width law and Rayleigh range.
Symbols & assumptions
Fundamental Gaussian beam in a uniform medium; λ is the wavelength in that medium and w₀ the beam waist.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. At a fixed axial plane, take a fundamental paraxial Gaussian beam with 1/e² intensity radius w.
I(r)/I(0)=e−2(r/w)2
Solution. Square the Gaussian field amplitude exp(−r²/w²) to obtain its intensity.
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Orange point: horizontal coordinate 0, calculated vertical coordinate 1. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 1 on the vertical axis. Values are rounded for display.
Scope. One transverse cut at a fixed plane; w changes with axial distance.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
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Represents free-carrier and bound-charge contributions to permittivity.
Cross-scalePhysical model
Mathematical model & short derivation
Representative formulation
ε(ω)=ε∞−ω2+iγωωp2+j∑ωj2−ω2−iγjωfj
Derivation / construction sketch
Model free carriers with a damped driven equation lacking a restoring force.
Model bound charges as damped driven oscillators.
Solve for polarization and add its contributions to permittivity.
Symbols & assumptions
Convention e^(−iωt); fj are oscillator strengths with compatible units. Parameters must be fitted to the relevant frequency range.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Practical applications & products
Application area
Electromagnetics & optics
Practical use
Frequency-dependent optical response of a material.
Problem & parameters. Take the free-electron Drude limit with zero collision rate, no Lorentz resonances, and background permittivity one.
ϵr=1−(ωp/ω)2
Solution. Solve the harmonic free-electron displacement equation and insert the induced polarization into D=ε0E+P.
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Orange point: horizontal coordinate 1.75, calculated vertical coordinate 0.67347. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 1.75, into the displayed formula to obtain 0.67347 on the vertical axis. Values are rounded for display.
Scope. Lossless frequency-domain special case; the zero-frequency singular point is excluded.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
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Uses resistors, capacitors and inductors connected by Kirchhoff laws.
Micro / mesoPhysical model
Mathematical model & short derivation
Representative formulation
Lq¨+Rq˙+q/C=Vin(t)i=q˙
Derivation / construction sketch
Apply Kirchhoff’s voltage law to a series resistor, inductor and capacitor.
Use vR=Ri, vL=Ldi/dt and vC=q/C.
Replace current by charge rate to obtain the second-order equation.
Symbols & assumptions
Representative series RLC circuit; lumped behavior assumes propagation delay is negligible.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Set inductance to zero and apply a voltage step Vs to a series resistor and initially uncharged capacitor.
VC/Vs=1−e−t/(RC)
Solution. Kirchhoff’s law gives RCV′+V=Vs. Solve the first-order initial-value problem.
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Orange point: horizontal coordinate 2.5, calculated vertical coordinate 0.91792. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.91792 on the vertical axis. Values are rounded for display.
Scope. RC limiting circuit, not a general second-order RLC transient.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
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Represents distributed inductance, capacitance and losses.
Micro / mesoPhysical model
Mathematical model & short derivation
Representative formulation
∂xV=−R′I−L′∂tI∂xI=−G′V−C′∂tV
Derivation / construction sketch
Represent a short line segment by series resistance/inductance and shunt conductance/capacitance.
Apply Kirchhoff laws.
Divide by segment length and take its limit to obtain the telegrapher equations.
Symbols & assumptions
R′,L′,G′,C′ are per-length parameters; frequency dependence and multiple conductors may require matrix forms.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Use a one-dimensional sinusoidal wave in a uniform, lossless linear medium. Plot the normalized field at time zero.
u(ξ,0)=sin(2πξ)
Solution. A sinusoidal traveling-wave solution is u = sin[2π(ξ−τ)]. Set τ = 0 to obtain the plotted snapshot.
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Orange point: horizontal coordinate 0.5, calculated vertical coordinate 1.2246e-16. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 1.2246e-16 on the vertical axis. Values are rounded for display.
Scope. An acoustic, electromagnetic, elastic, or linear Alfvén-wave reference as appropriate. For MHD this is the small transverse perturbation of a uniform magnetized equilibrium; for FDTD it is an exact target, not a discretized result.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
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Approximates diode current with an exponential voltage relation.
Micro / mesoPhysical model
Mathematical model & short derivation
Representative formulation
I=Is[exp(V/(nVT))−1]VT=qkBT
Derivation / construction sketch
Use the junction voltage to change minority-carrier concentrations exponentially.
Solve steady diffusion in the neutral regions.
Add electron and hole diffusion currents to obtain an exponential current law.
Symbols & assumptions
Ideal diffusion diode has n≈1; practical ideality factor n captures limited departures. Breakdown, series resistance and high injection need extensions.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Evaluate the Shockley diode law without series resistance or reverse breakdown.
I/Is=eV/(nVT)−1
Solution. Substitute the thermal-voltage-scaled bias into the exponential current law.
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Orange point: horizontal coordinate 0, calculated vertical coordinate 0. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 0 on the vertical axis. Values are rounded for display.
Scope. Analytical solution or constitutive evaluation for the stated special case. It is not a general solution of the full model family.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
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Models coupled junction currents in a bipolar transistor.
Micro / mesoPhysical model
Mathematical model & short derivation
Representative formulation
IC=αFIES(eVBE/VT−1)−ICS(eVBC/VT−1)
Derivation / construction sketch
Represent the emitter-base and collector-base junctions by coupled diode currents.
Transport a fraction αF of the forward emitter injection to the collector.
Subtract reverse collector-junction injection using a consistent terminal sign convention.
Symbols & assumptions
Representative NPN Ebers–Moll collector-current equation; companion emitter/base equations and reciprocity complete the model.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Example 1: Forward-active transistor collector current
Forward-active transistor collector current
Problem & parameters. Use the forward-active Ebers–Moll branch and neglect the reverse junction contribution.
IC/(αFIES)=eVBE/VT−1
Solution. Keep the αFIES[exp(VBE/VT)−1] term and divide by its prefactor.
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Orange point: horizontal coordinate 2, calculated vertical coordinate 6.3891. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 2, into the displayed formula to obtain 6.3891 on the vertical axis. Values are rounded for display.
Scope. Forward-active approximation; no saturation, Early effect, or breakdown.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
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Use the gradual-channel approximation to express local inversion charge.
Relate drift current to that charge and the channel voltage gradient.
Integrate along the channel; pinch-off yields the saturation expression.
Symbols & assumptions
Long-channel MOSFET, VGS>VT, negligible channel-length modulation, constant mobility. VT here means threshold voltage.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Use a long-channel MOSFET in strong-inversion saturation with constant mobility and no channel-length modulation.
ID/(βV∗2/2)=[(VGS−Vth)/V∗]2
Solution. Set VDS at or above overdrive and integrate the gradual-channel charge relation to obtain ID=β(VGS−Vth)²/2.
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Orange point: horizontal coordinate 1.5, calculated vertical coordinate 2.25. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 1.5, into the displayed formula to obtain 2.25 on the vertical axis. Values are rounded for display.
Scope. Analytical solution or constitutive evaluation for the stated special case. It is not a general solution of the full model family.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
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Start with channel charge and carrier transport physics.
Introduce calibrated corrections for short-channel, mobility, leakage and geometry effects.
Use consistent terminal charges to represent transient currents.
Symbols & assumptions
BSIM is a family of extensive compact models, not one universal formula. F and Gi denote the version-specific published equations and θ the process parameters.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Practical applications & products
Application area
Circuits & semiconductor devices
Practical use
Integrated-circuit simulation for a calibrated fabrication process.
Problem & parameters. Use an ideal weak-inversion exponential trend at fixed drain bias as a compact-model check.
ID/I∗=e(VGS−V∗)/(nVT)
Solution. A Boltzmann subthreshold charge law gives current proportional to exp(VGS/nVT); normalize at VGS=V*.
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Orange point: horizontal coordinate -1.5, calculated vertical coordinate 0.22313. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, -1.5, into the displayed formula to obtain 0.22313 on the vertical axis. Values are rounded for display.
Scope. Asymptotic benchmark only; not the complete BSIM equations or a result from a foundry model card.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
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Combines electrostatics with carrier drift, diffusion and continuity.
Micro / mesoPhysical model
Mathematical model & short derivation
Representative formulation
Jn=qμnnE+qDn∇nJp=qμppE−qDp∇p
Derivation / construction sketch
Combine carrier drift in the electric field with diffusion down concentration gradients.
Convert electron and hole particle fluxes to conventional charge currents.
Couple them to continuity and electrostatic Poisson equations.
Symbols & assumptions
n,p are carrier densities and q the positive elementary charge. Recombination, generation and boundary contacts must be specified.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Practical applications & products
Application area
Circuits & semiconductor devices
Practical use
Charge transport through a semiconductor junction.
Problem & parameters. Take uniform electron density n, fixed mobility μ, and a low-field steady state. The density gradient is zero.
J/(qnμE∗)=E/E∗
Solution. The diffusion contribution vanishes. Evaluate the conventional drift-current magnitude law J=qnμE.
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Orange point: horizontal coordinate 0, calculated vertical coordinate 0. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 0 on the vertical axis. Values are rounded for display.
Scope. Low-field isothermal drift limit; carrier heating and higher hydrodynamic moments are excluded.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
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Adds carrier-energy or momentum information to transport.
Micro / mesoPhysical model
Mathematical model & short derivation
Representative formulation
∂tWn+∇⋅SW=Jn⋅E−τEWn−Wn,eq
Derivation / construction sketch
Take an energy moment of the carrier Boltzmann equation.
Represent field work as Jn·E.
Close the energy flux and scattering loss using a carrier-energy relaxation approximation.
Symbols & assumptions
Wn is carrier energy density, SW energy flux and τE relaxation time; hydrodynamic variants add momentum moments and different closures.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Take uniform electron density n, fixed mobility μ, and a low-field steady state. The density gradient is zero.
J/(qnμE∗)=E/E∗
Solution. The diffusion contribution vanishes. Evaluate the conventional drift-current magnitude law J=qnμE.
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Orange point: horizontal coordinate 0, calculated vertical coordinate 0. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 0 on the vertical axis. Values are rounded for display.
Scope. Low-field isothermal drift limit; carrier heating and higher hydrodynamic moments are excluded.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
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Relates electrochemical equilibrium potential to species activities.
Component / systemPhysical model
Mathematical model & short derivation
Representative formulation
E=E∘−nFRTlnQ
Derivation / construction sketch
Write reaction Gibbs energy as ΔG=ΔG°+RTln Q.
Relate reversible electrical work to −nFE.
Combine the two expressions and define E°=−ΔG°/(nF).
Symbols & assumptions
Q is the activity-based reaction quotient, n transferred electrons and F Faraday’s constant. Equilibrium is required.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Example 1: Equilibrium potential versus activity ratio
Equilibrium potential versus activity ratio
Problem & parameters. For Ox+ne− ⇌ Red use ideal specified activities and fixed temperature.
nF(E−E∘)/(RT)=ln(aox/ared)
Solution. Set the reaction electrochemical free-energy change to zero and rearrange the Nernst relation.
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Orange point: horizontal coordinate 5.05, calculated vertical coordinate 1.6194. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 5.05, into the displayed formula to obtain 1.6194 on the vertical axis. Values are rounded for display.
Scope. Analytical solution or constitutive evaluation for the stated special case. It is not a general solution of the full model family.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
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Relates interfacial current to electrochemical overpotential.
Component / systemPhysical model
Mathematical model & short derivation
Representative formulation
j=j0[exp(αaFη/(RT))−exp(−αcFη/(RT))]
Derivation / construction sketch
Treat anodic and cathodic reaction rates as activated processes.
Let overpotential η shift the forward and reverse activation barriers.
Subtract the two partial currents and require zero net current at equilibrium.
Symbols & assumptions
One-electron notation shown; stoichiometric and transfer-coefficient conventions must match the reaction mechanism.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Set anodic and cathodic transfer coefficients to one half, with one-electron charge convention.
j/j0=2sinh(η∗/2),η∗=Fη/(RT)
Solution. Subtract the two exponentials in Butler–Volmer to obtain twice the hyperbolic sine.
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Orange point: horizontal coordinate 0, calculated vertical coordinate 0. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 0 on the vertical axis. Values are rounded for display.
Scope. Analytical solution or constitutive evaluation for the stated special case. It is not a general solution of the full model family.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
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Approximates high-overpotential behavior of Butler–Volmer kinetics.
Component / systemPhysical model
Mathematical model & short derivation
Representative formulation
η≈αaFRTln(j/j0)
Derivation / construction sketch
Start with Butler–Volmer kinetics.
At sufficiently large positive overpotential, neglect the cathodic exponential.
Take the logarithm to obtain a straight-line Tafel relation.
Symbols & assumptions
Anodic branch shown; mass-transfer limits, ohmic losses and surface changes must be separated from activation kinetics.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Practical applications & products
Application area
Electrochemistry & energy storage
Practical use
Interpreting a polarization curve in a suitable regime.
Problem & parameters. Use the anodic high-overpotential regime where the cathodic exponential is negligible.
αFη/(RT)=ln(j/j0)
Solution. From j≈j0 exp(αFη/RT), take logarithms and solve for overpotential.
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Orange point: horizontal coordinate 55, calculated vertical coordinate 4.0073. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 55, into the displayed formula to obtain 4.0073 on the vertical axis. Values are rounded for display.
Scope. Asymptotic approximation, plotted well above j/j0=1; not valid near equilibrium.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
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Couples electrostatics to diffusion and migration of ions.
Component / systemPhysical model
Mathematical model & short derivation
Representative formulation
Ji=−Di[∇ci+RTziFci∇ϕ]−∇⋅(ε∇ϕ)=Fi∑zici
Derivation / construction sketch
Combine diffusion with electric-field-driven ion migration using the Einstein relation.
Use ion conservation ∂tci=−∇·Ji plus reactions if present.
Determine electric potential from the local ionic charge density.
Symbols & assumptions
Dilute continuum electrolyte without advection shown; concentrated electrolytes and ion correlations require richer constitutive laws.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Example 1: Screened potential in a dilute electrolyte
Screened potential in a dilute electrolyte
Problem & parameters. At zero ionic flux, linearize a symmetric dilute electrolyte near equilibrium next to a planar wall.
ϕ/ϕ0=e−x/λD
Solution. Boltzmann ionic populations linearize Poisson’s equation to φ″=φ/λD². Select the decaying solution and impose the wall potential.
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Orange point: horizontal coordinate 2.5, calculated vertical coordinate 0.082085. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.082085 on the vertical axis. Values are rounded for display.
Scope. Debye–Hückel equilibrium limit of PNP, requiring |zFφ|≪RT; no driven ionic transport.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
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Couple reaction fluxes to porous-electrode electrolyte and electronic transport along cell thickness x.
Enforce current conservation and Butler–Volmer interfacial kinetics.
Symbols & assumptions
Schematic DFN/P2D core; electrolyte mass balance, potential equations, porosity factors and boundary conditions complete the model. BV denotes Butler–Volmer kinetics.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Example 1: Spherical-particle average concentration balance
Spherical-particle average concentration balance
Problem & parameters. Start with a spherical active particle of radius R and mean concentration c*. Impose constant outward molar flux jout.
cˉ/c∗=1−3τ,τ=joutt/(Rc∗)
Solution. Integrate spherical diffusion over particle volume: d(c̄)/dt=−(surface/volume)jout=−3jout/R. Apply the initial average.
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Orange point: horizontal coordinate 0.125, calculated vertical coordinate 0.625. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.125, into the displayed formula to obtain 0.625 on the vertical axis. Values are rounded for display.
Scope. Exact particle mass balance shared by DFN, SPM, and SPMe. It does not give the radial profile, terminal voltage, electrolyte dynamics, or a usable-capacity prediction.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
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Replace each porous electrode’s particle population by one representative particle.
Apply the average reaction flux implied by cell current.
Use surface concentrations to compute equilibrium potentials and kinetic voltage losses.
Symbols & assumptions
SPM simplification; electrolyte concentration and potential variations are neglected or approximated.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Example 1: Spherical-particle average concentration balance
Spherical-particle average concentration balance
Problem & parameters. Start with a spherical active particle of radius R and mean concentration c*. Impose constant outward molar flux jout.
cˉ/c∗=1−3τ,τ=joutt/(Rc∗)
Solution. Integrate spherical diffusion over particle volume: d(c̄)/dt=−(surface/volume)jout=−3jout/R. Apply the initial average.
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Orange point: horizontal coordinate 0.125, calculated vertical coordinate 0.625. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.125, into the displayed formula to obtain 0.625 on the vertical axis. Values are rounded for display.
Scope. Exact particle mass balance shared by DFN, SPM, and SPMe. It does not give the radial profile, terminal voltage, electrolyte dynamics, or a usable-capacity prediction.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Adds electrolyte concentration effects to a single-particle approximation.
Component / systemPhysical model
Mathematical model & short derivation
Representative formulation
εe∂tce=∂x(De,eff∂xce)+(1−t+)aj/F
Derivation / construction sketch
Retain the SPM particle equations.
Add electrolyte salt conservation across electrode and separator regions.
Use electrolyte concentration and potential corrections in the cell voltage.
Symbols & assumptions
Representative electrolyte balance within SPMe; j is interfacial current density and a interfacial area per volume, with region-dependent signs and coefficients.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Example 1: Spherical-particle average concentration balance
Spherical-particle average concentration balance
Problem & parameters. Start with a spherical active particle of radius R and mean concentration c*. Impose constant outward molar flux jout.
cˉ/c∗=1−3τ,τ=joutt/(Rc∗)
Solution. Integrate spherical diffusion over particle volume: d(c̄)/dt=−(surface/volume)jout=−3jout/R. Apply the initial average.
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Orange point: horizontal coordinate 0.125, calculated vertical coordinate 0.625. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.125, into the displayed formula to obtain 0.625 on the vertical axis. Values are rounded for display.
Scope. Exact particle mass balance shared by DFN, SPM, and SPMe. It does not give the radial profile, terminal voltage, electrolyte dynamics, or a usable-capacity prediction.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Uses fitted electrical elements to approximate terminal behavior.
Component / systemPhysical model
Mathematical model & short derivation
Representative formulation
V=OCV(z)−IR0−v1v˙1=−R1C1v1+C1Iz˙=−QnI
Derivation / construction sketch
Represent instantaneous ohmic loss by R₀ and relaxation by an RC branch.
Apply Kirchhoff’s laws to the branch.
Track state of charge z by coulomb counting.
Symbols & assumptions
One-RC Thevenin battery model with discharge-positive current; parameters depend on temperature, charge state and aging.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Example 1: Battery polarization under a current step
Battery polarization under a current step
Problem & parameters. Apply a constant current I to an initially relaxed single-RC battery polarization branch.
Vp/(IRp)=1−e−t/(RpCp)
Solution. Solve CpVp′+Vp/Rp=I. The terminal-voltage drop also includes any separate series ohmic resistance.
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Orange point: horizontal coordinate 2.5, calculated vertical coordinate 0.91792. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.91792 on the vertical axis. Values are rounded for display.
Scope. One branch with fixed parameters; state of charge and open-circuit voltage are held fixed.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Relates averaged fluid flux to hydraulic gradient.
Continuum / componentPhysical model
Mathematical model & short derivation
Representative formulation
q=−μK(∇p−ρg)
Derivation / construction sketch
Average slow viscous flow over a representative porous volume.
Relate bulk flux linearly to pressure and gravity driving forces.
Collect pore-geometry effects into permeability K.
Symbols & assumptions
q is Darcy volumetric flux, not pore velocity; K may be a tensor and μ is dynamic viscosity.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Let G=−dp/dx be positive, and hold permeability k and viscosity μ constant.
u/(kG∗/μ)=G/G∗
Solution. Solve μu/k = G.
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Orange point: horizontal coordinate 1.5, calculated vertical coordinate 1.5. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 1.5, into the displayed formula to obtain 1.5 on the vertical axis. Values are rounded for display.
Scope. Single-phase creeping flow in a homogeneous porous medium.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
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Adds a viscous shear term to a Darcy-like resistance model.
Continuum / componentPhysical model
Mathematical model & short derivation
Representative formulation
−∇p+μeff∇2u−μK−1u+ρg=0
Derivation / construction sketch
Begin with Darcy drag in a homogenized porous medium.
Add a viscous shear-diffusion term to represent momentum exchange across velocity gradients.
Balance pressure, shear, porous resistance and body force.
Symbols & assumptions
Brinkman effective viscosity μeff is model-dependent; interface conditions between porous and free flow require care.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Solve μe u″−μu/k+G=0 between no-slip walls ±H. Choose screening length ℓ=√(μe k/μ) and H/ℓ=2.
u/(kG/μ)=1−cosh(x/ℓ)/cosh(H/ℓ),H/ℓ=2
Solution. Add a constant particular solution kG/μ to the symmetric cosh homogeneous solution, then enforce the wall values.
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Orange point: horizontal coordinate 0, calculated vertical coordinate 0.7342. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 0.7342 on the vertical axis. Values are rounded for display.
Scope. Analytical solution or constitutive evaluation for the stated special case. It is not a general solution of the full model family.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
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Start with linear Darcy resistance at small pore Reynolds number.
Add a quadratic velocity-dependent inertial loss.
Fit the coefficient β to porous geometry or measurements.
Symbols & assumptions
Isotropic form without gravity; β has inverse-length units. Velocity convention must match calibration.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Choose velocity and gradient scales so that the linear and quadratic drag coefficients are both one.
G/G∗=v+v2
Solution. Substitute positive velocity into G=av+bv|v| and apply the chosen scaling.
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Orange point: horizontal coordinate 1.5, calculated vertical coordinate 3.75. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 1.5, into the displayed formula to obtain 3.75 on the vertical axis. Values are rounded for display.
Scope. Analytical solution or constitutive evaluation for the stated special case. It is not a general solution of the full model family.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
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Describes variably saturated water movement in porous media.
Continuum / componentPhysical model
Mathematical model & short derivation
Representative formulation
∂t∂θ(h)=∇⋅[K(h)∇(h+z)]
Derivation / construction sketch
Apply water conservation to a variably saturated porous medium.
Use a saturation-dependent Darcy flux driven by pressure head h plus elevation z.
Close water content θ and hydraulic conductivity K as functions of h.
Symbols & assumptions
Water phase only with air pressure approximated as known; hysteresis and preferential flow are omitted unless added.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Linearize moisture capacity and hydraulic conductivity about a uniform reference state, neglect gravity, and solve the resulting diffusion equation on a slab.
u(ξ,τ)=sin(πξ)e−π2τ,τ=0.1
Solution. The sine satisfies both zero end values. Its second derivative is −π² times itself; the amplitude solves a′ = −π²a.
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Orange point: horizontal coordinate 0.5, calculated vertical coordinate 0.37271. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 0.37271 on the vertical axis. Values are rounded for display.
Scope. Constant-coefficient linearization of Richards’ equation. The nonlinear retention and conductivity changes are excluded.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
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Relates water saturation to pressure head with fitted parameters.
Continuum / componentPhysical model
Mathematical model & short derivation
Representative formulation
Se=[1+(α∣h∣)n]−mθ=θr+Se(θs−θr)
Derivation / construction sketch
Normalize water content between residual and saturated limits.
Choose a monotonic fitted function of suction head.
Use parameters α,n,m to capture the observed retention curve.
Symbols & assumptions
For h<0; saturated branch has Se=1. Often m=1−1/n for a particular conductivity closure, but this is not mandatory for the retention relation alone.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Choose n=2 and m=1−1/n=1/2 for a drying retention curve.
Se=[1+(α∣h∣)2]−1/2
Solution. Insert the chosen parameters into Se=[1+(α|h|)^n]^−m.
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Orange point: horizontal coordinate 2.5, calculated vertical coordinate 0.37139. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.37139 on the vertical axis. Values are rounded for display.
Scope. Retention relation only; hysteresis and conductivity are not evaluated.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
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Couples solid deformation and pore-fluid pressure.
Continuum / componentPhysical model
Mathematical model & short derivation
Representative formulation
σ=C:ε−αBpIζ=αBtrε+p/Mζ˙+∇⋅q=0
Derivation / construction sketch
Split total stress into skeleton deformation and pore-pressure contributions.
Relate fluid-content change ζ to volumetric strain and pressure.
Combine fluid conservation with Darcy flow and mechanical equilibrium.
Symbols & assumptions
Linear Biot poroelasticity; αB is Biot coefficient and M Biot modulus. Sign conventions must match the strain and stress definitions.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Use one-dimensional linear consolidation with drained ends and an initial excess pore-pressure mode sin(πx/L). Plot cvt/L²=0.1.
u(ξ,τ)=sin(πξ)e−π2τ,τ=0.1
Solution. The sine satisfies both zero end values. Its second derivative is −π² times itself; the amplitude solves a′ = −π²a.
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Orange point: horizontal coordinate 0.5, calculated vertical coordinate 0.37271. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 0.37271 on the vertical axis. Values are rounded for display.
Scope. Exact single-mode Terzaghi solution and a compatible one-dimensional poroelastic reduction; not an arbitrary initial loading history.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Describes time-dependent settlement from pore-pressure dissipation.
Continuum / componentPhysical model
Mathematical model & short derivation
Representative formulation
∂t∂u=cv∂z2∂2ucv=mvγwk
Derivation / construction sketch
Combine one-dimensional fluid conservation with Darcy drainage.
Relate volume change to effective-stress change using compressibility mv.
For constant total load, eliminate strain to obtain pore-pressure diffusion.
Symbols & assumptions
u is excess pore pressure, k hydraulic conductivity and γw water unit weight; assumptions include saturated homogeneous soil and small strain.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Use one-dimensional linear consolidation with drained ends and an initial excess pore-pressure mode sin(πx/L). Plot cvt/L²=0.1.
u(ξ,τ)=sin(πξ)e−π2τ,τ=0.1
Solution. The sine satisfies both zero end values. Its second derivative is −π² times itself; the amplitude solves a′ = −π²a.
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Orange point: horizontal coordinate 0.5, calculated vertical coordinate 0.37271. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 0.37271 on the vertical axis. Values are rounded for display.
Scope. Exact single-mode Terzaghi solution and a compatible one-dimensional poroelastic reduction; not an arbitrary initial loading history.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Uses critical-state plasticity for idealized clay behavior.
Continuum / componentPhysical model
Mathematical model & short derivation
Representative formulation
f=q2+M2p′(p′−pc′)=0
Derivation / construction sketch
Describe yielding in mean effective stress p′ and deviatoric stress q.
Use an elliptical surface that meets the critical-state line q=Mp′.
Evolve preconsolidation pressure pc′ through a volumetric hardening law.
Symbols & assumptions
Modified Cam-Clay with compression-positive stress; elasticity, associated flow and hardening complete the constitutive model.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Hold preconsolidation pressure pc and critical-state slope M fixed. Plot the compression-positive yield locus.
q/(Mpc)=(p/pc)(1−p/pc)
Solution. Solve q²+M²p(p−pc)=0 for the nonnegative q branch.
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Orange point: horizontal coordinate 0.5, calculated vertical coordinate 0.5. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 0.5 on the vertical axis. Values are rounded for display.
Scope. Yield-surface geometry only; hardening and stress-path evolution are not solved.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
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Integrate incompressible conservation through water depth.
Assume vertical acceleration is small so pressure is hydrostatic.
Represent bed slope and friction as depth-averaged source terms.
Symbols & assumptions
Two-dimensional shallow-water form; h is depth and zb bed elevation. Waves must be long relative to depth.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Linearize shallow-water dynamics about rest at constant depth H; the wave speed is √(gH).
u(ξ,0)=sin(2πξ)
Solution. A sinusoidal traveling-wave solution is u = sin[2π(ξ−τ)]. Set τ = 0 to obtain the plotted snapshot.
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Orange point: horizontal coordinate 0.5, calculated vertical coordinate 1.2246e-16. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 1.2246e-16 on the vertical axis. Values are rounded for display.
Scope. Small free-surface perturbation in a constant-depth channel, without friction or dispersion.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
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Simplifies flow routing by approximating dominant slope and friction balance.
Regional / planetaryPhysical model
Mathematical model & short derivation
Representative formulation
∂t∂A+∂x∂Q=qlQ=αAm
Derivation / construction sketch
Keep cross-sectional mass conservation.
Approximate momentum by local friction-slope balance.
Use an algebraic discharge-area relation to close the routing equation.
Symbols & assumptions
A is wetted area and ql lateral inflow per length; backwater and inertia are poorly represented.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Use the linear routing equation ht+hx=0 with initial Gaussian pulse exp(−x²).
h(x,1)=e−(x−1)2
Solution. The pulse is constant along characteristics x−t, hence it translates without changing shape.
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Orange point: horizontal coordinate 1, calculated vertical coordinate 1. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 1, into the displayed formula to obtain 1 on the vertical axis. Values are rounded for display.
Scope. Constant-celerity reduction; nonlinear depth-dependent routing can distort or steepen the pulse.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
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Converts precipitation and catchment storage into streamflow.
Regional / planetaryPhysical model
Mathematical model & short derivation
Representative formulation
dtdS=P−ET−QQ=f(S,soil,routing)
Derivation / construction sketch
Apply catchment water balance.
Partition rainfall into storage, evapotranspiration and outflow.
Specify empirical or physical functions for infiltration, storage release and channel routing.
Symbols & assumptions
Representative rainfall–runoff structure; there is no single universal model. Spatial resolution and parameter choices define a particular implementation.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. After rainfall stops, let storage S obey S′=−S/K and outflow Q=S/K. Normalize either by its initial value.
y(τ)=e−τ
Solution. Separate dy/dτ = −y, integrate ln(y) = −τ + C, and apply y(0) = 1.
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Orange point: horizontal coordinate 2.5, calculated vertical coordinate 0.082085. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.082085 on the vertical axis. Values are rounded for display.
Scope. One-reservoir rainfall–runoff component; no new rain, infiltration, or additional routing stores.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
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Combines water conservation with porous-flow relations.
Regional / planetaryPhysical model
Mathematical model & short derivation
Representative formulation
Ss∂t∂h=∇⋅(K∇h)+W
Derivation / construction sketch
Apply water conservation in a saturated porous volume.
Insert Darcy flux q=−K∇h.
Represent compressible storage with specific storage Ss and sources with W.
Symbols & assumptions
Hydraulic-head form for saturated flow; unconfined aquifers require appropriate water-table storage and moving-boundary treatment.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Example 1: Steady one-dimensional diffusion benchmark
Steady one-dimensional diffusion benchmark
Problem & parameters. Solve u″ = 0 on 0 < ξ < 1 with u(0) = 1 and u(1) = 0, constant transport coefficient, and no source.
u(ξ)=1−ξ
Solution. Integrate twice to obtain u = A+Bξ. The two endpoint values give A = 1 and B = −1.
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Orange point: horizontal coordinate 0.5, calculated vertical coordinate 0.5. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 0.5 on the vertical axis. Values are rounded for display.
Scope. For numerical-method entries this is the exact target to verify against, not a computed discretization or convergence claim.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
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Represents contaminant transport and spreading through an aquifer.
Regional / planetaryPhysical model
Mathematical model & short derivation
Representative formulation
∂t∂(θc)=∇⋅(θD∇c)−∇⋅(qc)+S
Derivation / construction sketch
Balance contaminant mass in pore water.
Represent bulk transport by Darcy flux q and spreading by a dispersion tensor D.
Add sources, reactions and sorption storage as needed.
Symbols & assumptions
θ is porosity or water content; the written storage term omits sorbed mass, which must be added for retarding solutes.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. On an infinite line take velocity one, dispersion coefficient 0.1, and initial concentration exp(−x²).
c(x,1)=1.41e−(x−1)2/1.4
Solution. Translate the Gaussian by vt and broaden its squared width to 1+4Dt, adjusting amplitude to conserve mass.
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Orange point: horizontal coordinate 1, calculated vertical coordinate 0.84515. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 1, into the displayed formula to obtain 0.84515 on the vertical axis. Values are rounded for display.
Scope. Homogeneous advection–dispersion with no reactions or sorption.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
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Evolves atmospheric dynamics and thermodynamics from an analyzed initial state.
Regional / planetaryPhysical model
Mathematical model & short derivation
Representative formulation
DtDu+2Ω×u=−ρ∇p+g+FDtDθ=Qθ
Derivation / construction sketch
Apply rotating-frame momentum conservation.
Couple it to mass, thermodynamic and water-species balances.
Discretize, initialize from observations and parameterize unresolved processes to produce forecasts.
Symbols & assumptions
Schematic atmospheric dynamics; θ is potential temperature, Ω Earth’s rotation. Hydrostatic versus nonhydrostatic formulations differ.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Use an ideal gas at constant temperature and constant gravity, with density ρ0 at height zero.
ρ(z)/ρ0=e−z/H
Solution. Combine dp/dz=−ρg with p=ρRsT; integrate dρ/dz=−ρ/H, H=RsT/g.
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Orange point: horizontal coordinate 2.5, calculated vertical coordinate 0.082085. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.082085 on the vertical axis. Values are rounded for display.
Scope. Hydrostatic column benchmark only. For stellar structure this approximates a thin isothermal layer, not an entire star; radiation, convection, and dynamics are excluded.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
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Represents large-scale atmospheric or oceanic circulation.
Regional / planetaryPhysical model
Mathematical model & short derivation
Representative formulation
∂tx=Fdyn(x)+Fphysics(x,forcing)
Derivation / construction sketch
Represent the discretized atmosphere or ocean by state vector x.
Advance resolved conservation laws with dynamical operator Fdyn.
Add radiation, mixing, cloud or other unresolved physical tendencies.
Symbols & assumptions
GCM is a model class rather than one equation; spatial discretization, coupling and parameterizations specify the actual model.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Practical applications & products
Application area
Water, atmosphere & Earth systems
Practical use
Studying global circulation under specified forcing.
Problem & parameters. Use an ideal gas at constant temperature and constant gravity, with density ρ0 at height zero.
ρ(z)/ρ0=e−z/H
Solution. Combine dp/dz=−ρg with p=ρRsT; integrate dρ/dz=−ρ/H, H=RsT/g.
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Orange point: horizontal coordinate 2.5, calculated vertical coordinate 0.082085. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.082085 on the vertical axis. Values are rounded for display.
Scope. Hydrostatic column benchmark only. For stellar structure this approximates a thin isothermal layer, not an entire star; radiation, convection, and dynamics are excluded.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Build separate atmosphere, ocean and land evolution models.
Exchange heat, water, momentum and biogeochemical fluxes across their boundaries.
Enforce compatible time stepping and conservation during coupling.
Symbols & assumptions
Schematic Earth-system coupling; subscripts identify components and exchanged fluxes, with ice and chemistry often added.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Practical applications & products
Application area
Water, atmosphere & Earth systems
Practical use
Exploring climate responses to emissions scenarios.
Example 1: One-box climate response to a forcing step
One-box climate response to a forcing step
Problem & parameters. For a constant radiative-forcing step F, use CΔT′=F−λΔT with positive linear feedback parameter λ and initially zero anomaly.
ΔT/(F/λ)=1−e−λt/C
Solution. Apply an integrating factor to the one-box energy balance; the equilibrium anomaly is F/λ.
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Orange point: horizontal coordinate 2.5, calculated vertical coordinate 0.91792. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.91792 on the vertical axis. Values are rounded for display.
Scope. Reduced global-mean energy balance. For ESM this is an illustrative diagnostic reduction, not a full Earth-system forecast.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Balances incoming and outgoing energy in a simplified climate system.
Regional / planetaryPhysical model
Mathematical model & short derivation
Representative formulation
CdtdT=4(1−α)S−OLR(T)
Derivation / construction sketch
Average absorbed sunlight over the planetary surface.
Subtract outgoing longwave radiation OLR.
Assign the residual to heat storage C dT/dt.
Symbols & assumptions
Zero-dimensional energy-balance climate model; C is heat capacity per area, S solar irradiance and α planetary albedo.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Practical applications & products
Application area
Water, atmosphere & Earth systems
Practical use
Estimating idealized temperature response to forcing.
Example 1: One-box climate response to a forcing step
One-box climate response to a forcing step
Problem & parameters. For a constant radiative-forcing step F, use CΔT′=F−λΔT with positive linear feedback parameter λ and initially zero anomaly.
ΔT/(F/λ)=1−e−λt/C
Solution. Apply an integrating factor to the one-box energy balance; the equilibrium anomaly is F/λ.
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Orange point: horizontal coordinate 2.5, calculated vertical coordinate 0.91792. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.91792 on the vertical axis. Values are rounded for display.
Scope. Reduced global-mean energy balance. For ESM this is an illustrative diagnostic reduction, not a full Earth-system forecast.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Apply rotating-fluid momentum and mass conservation.
Use the Boussinesq approximation and hydrostatic vertical balance for large-scale flow.
Couple velocity to temperature and salinity transport and an equation of state.
Symbols & assumptions
Representative primitive-equation ocean model; nonhydrostatic effects matter at smaller scales.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Use a constant-depth, nonrotating, inviscid shallow-water reduction of ocean circulation.
u(ξ,0)=sin(2πξ)
Solution. A sinusoidal traveling-wave solution is u = sin[2π(ξ−τ)]. Set τ = 0 to obtain the plotted snapshot.
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Orange point: horizontal coordinate 0.5, calculated vertical coordinate 1.2246e-16. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 1.2246e-16 on the vertical axis. Values are rounded for display.
Scope. Single linear barotropic mode; rotation, stratification, mixing, and realistic boundaries are excluded.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Balance sea-ice volume through transport, freezing and melting.
Balance ice momentum with surface forcing and internal stress.
Close conductive heat flow and an ice rheology to couple thickness and motion.
Symbols & assumptions
h is ice thickness and mi mass per area; concentration, ridging and rheological formulations vary by model.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Assume zero initial thickness, fixed surface-to-freezing temperature difference ΔT, and conductive flux kΔT/h through the ice.
h/ℓ=t/t∗
Solution. Balance latent heat: ρLh′=kΔT/h. Integrate h²=2kΔTt/(ρL) and choose t*=ρLℓ²/(2kΔT).
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Orange point: horizontal coordinate 2, calculated vertical coordinate 1.4142. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 2, into the displayed formula to obtain 1.4142 on the vertical axis. Values are rounded for display.
Scope. Stefan growth limit with no ocean heat flux, snow insulation, or ice dynamics.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Propagates elastic disturbances through Earth materials.
Regional / planetaryPhysical model
Mathematical model & short derivation
Representative formulation
ρu¨=∇⋅σ+fσ=C:ε(u)
Derivation / construction sketch
Apply momentum conservation to an elastic solid.
Use a constitutive relation between stress and displacement gradients.
Propagate the resulting wave equation through heterogeneous Earth materials.
Symbols & assumptions
Linear elastic seismic model; attenuation, anisotropy, free-surface and absorbing boundaries may be required.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Practical applications & products
Application area
Water, atmosphere & Earth systems
Practical use
Ground-motion simulation for an earthquake scenario.
Problem & parameters. Use a one-dimensional sinusoidal wave in a uniform, lossless linear medium. Plot the normalized field at time zero.
u(ξ,0)=sin(2πξ)
Solution. A sinusoidal traveling-wave solution is u = sin[2π(ξ−τ)]. Set τ = 0 to obtain the plotted snapshot.
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Orange point: horizontal coordinate 0.5, calculated vertical coordinate 1.2246e-16. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 1.2246e-16 on the vertical axis. Values are rounded for display.
Scope. An acoustic, electromagnetic, elastic, or linear Alfvén-wave reference as appropriate. For MHD this is the small transverse perturbation of a uniform magnetized equilibrium; for FDTD it is an exact target, not a discretized result.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Evolves vehicle translation and rotation using aerodynamic and propulsion forces.
Component / systemPhysical model
Mathematical model & short derivation
Representative formulation
mv˙body+ω×(mvbody)=FIω˙+ω×Iω=M
Derivation / construction sketch
Resolve translational and rotational momentum in vehicle-fixed axes.
Include rotating-coordinate transport terms.
Compute aerodynamic, thrust and gravity loads and integrate attitude and position kinematics.
Symbols & assumptions
Six-degree-of-freedom rigid vehicle model; aerodynamic coefficients and mass properties are operating-condition dependent.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. A rigid body starts at rest with constant net force-to-mass ratio 1 m/s² along one axis and zero net torque.
x(t)=21at2,a=1m/s2
Solution. Newton’s law gives constant acceleration. Integrate twice with zero initial position and velocity.
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Orange point: horizontal coordinate 2.5, calculated vertical coordinate 3.125. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 3.125 on the vertical axis. Values are rounded for display.
Scope. Single translational degree of freedom; the remaining forces, torques, and rotational motion are set to zero.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Use lifting-line theory for an ideal elliptically loaded wing of aspect ratio eight and two-dimensional slope 2π per radian.
CL=1+2/(eAR)2πα,e=1,AR=8
Solution. The induced angle reduces the effective angle. Solve CL=a0[α−CL/(πeAR)] for CL.
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Orange point: horizontal coordinate 0, calculated vertical coordinate 0. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 0 on the vertical axis. Values are rounded for display.
Scope. Small-angle attached-flow approximation; no stall prediction.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Combines blade-section loads with momentum balances.
Component / systemPhysical model
Mathematical model & short derivation
Representative formulation
dT=21ρW2BcCl,normaldr=4πρU∞2a(1−a)rdr
Derivation / construction sketch
Compute blade-section loads from relative speed W, chord c and sectional coefficients.
Compute the same annular thrust from axial momentum theory.
Equate the two and iterate for induction factors.
Symbols & assumptions
Representative axial BEM balance; B is blade count and a axial induction. Tip loss, swirl and high-induction corrections are important.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Use the ideal nonrotating actuator-disk limit underlying axial momentum theory.
CP=4a(1−a)2
Solution. Mass, momentum, and energy balances give the displayed coefficient. Differentiating yields a maximum 16/27 at a=1/3.
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Orange point: horizontal coordinate 0.25, calculated vertical coordinate 0.5625. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.25, into the displayed formula to obtain 0.5625 on the vertical axis. Values are rounded for display.
Scope. Momentum-theory benchmark for BEM; blade geometry, swirl, drag, tip losses, and high-induction corrections are excluded.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Combines left and right wheels into a planar steering model.
Component / systemPhysical model
Mathematical model & short derivation
Representative formulation
m(v˙y+Ur)=Fyf+FyrIzr˙=lfFyf−lrFyr
Derivation / construction sketch
Merge left and right wheels into one front and one rear tire.
Apply planar lateral-force and yaw-moment balances.
Relate tire forces to slip angles, often through linear cornering stiffness.
Symbols & assumptions
Small-slip constant-forward-speed bicycle model; U is forward speed, r yaw rate and lf,lr axle distances from the mass center.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Assume low-speed rolling without tire slip for a vehicle of wheelbase L.
κL=tanδ
Solution. The front-wheel geometry gives turn radius R=L/tanδ; curvature is 1/R.
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Orange point: horizontal coordinate 0, calculated vertical coordinate 0. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 0 on the vertical axis. Values are rounded for display.
Scope. Kinematic limit, not a high-speed dynamic tire-force model.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Represent a quarter body and wheel assembly by sprung and unsprung masses.
Connect them with suspension stiffness and damping and connect the wheel to the road through tire stiffness.
Apply vertical force balance to each mass.
Symbols & assumptions
Linear two-mass quarter-car model; road displacement zr is input, with tire damping and active force omitted here.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Hold the unsprung mass fixed, set damping to zero, and release the sprung mass from displacement A.
z/A=cos(ωt)
Solution. The reduced quarter-car equation is ms z″+ks z=0; ω=√(ks/ms).
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Orange point: horizontal coordinate 6.2832, calculated vertical coordinate 1. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 6.2832, into the displayed formula to obtain 1 on the vertical axis. Values are rounded for display.
Scope. Single-mode constrained reduction, not the full two-degree-of-freedom road response.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Uses empirical nonlinear formulas for tire forces.
Component / systemPhysical model
Mathematical model & short derivation
Representative formulation
F=Dsin[Carctan(Bs−E(Bs−arctanBs))]
Derivation / construction sketch
Choose a flexible empirical curve with tunable initial slope, peak and curvature.
Fit its coefficients to tire-force measurements as functions of load and other conditions.
Evaluate the fitted force against slip s.
Symbols & assumptions
Simplified Pacejka Magic Formula; B,C,D,E are fitted coefficients, not universal constants. Combined slip and camber require extensions.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Choose B=10, C=1.3, E=0, zero offsets, and fixed load in the basic Pacejka Magic Formula.
F/D=sin[1.3arctan(10s)]
Solution. With E=0 the curvature correction drops out. Evaluate the sine of the scaled arctangent.
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Orange point: horizontal coordinate 0, calculated vertical coordinate 0. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 0 on the vertical axis. Values are rounded for display.
Scope. Illustrative coefficients, not a calibrated tire or a combined-slip model.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Write nodal current from the network admittance matrix: I=YV.
Use complex power S=VI*.
Separate real and imaginary equations and solve for unknown voltage magnitudes and angles.
Symbols & assumptions
AC steady-state phasor model; specified loads, generators, transformers and bus types complete the nonlinear system.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Use two fixed voltage magnitudes connected by a purely reactive line; for a generator use the analogous fixed internal-voltage coupling.
P/(V1V2/X)=sinδ
Solution. The lossless AC circuit gives P=(V1V2/X)sinδ.
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Orange point: horizontal coordinate 0, calculated vertical coordinate 0. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 0 on the vertical axis. Values are rounded for display.
Scope. Steady electrical-power term; the swing-equation rotor transient and voltage dynamics are not solved.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Linearizes active-power flow under restrictive grid assumptions.
Component / systemPhysical model
Mathematical model & short derivation
Representative formulation
Pij≈xijθi−θjP=Bθ
Derivation / construction sketch
Start from AC power-flow relations.
Assume near-unit voltage magnitudes, small angle differences and negligible resistance.
Linearize the sine term to obtain a susceptance-based active-power model.
Symbols & assumptions
Despite its name, this approximates an AC network; it does not represent reactive power, voltage magnitude changes or most losses.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Use nearly equal fixed bus voltage magnitudes, negligible resistance, and small angle difference.
P/(V1V2/X)≈δ
Solution. Linearize sinδ≈δ in the lossless AC transfer formula.
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Orange point: horizontal coordinate 0, calculated vertical coordinate 0. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 0 on the vertical axis. Values are rounded for display.
Scope. DC power-flow approximation; it does not calculate reactive power or voltage magnitudes.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Represents rotor-angle dynamics from mechanical-electrical power imbalance.
Component / systemPhysical model
Mathematical model & short derivation
Representative formulation
Mδ¨+Dδ˙=Pm−Pe(δ)
Derivation / construction sketch
Balance mechanical and electrical torque on a synchronous rotor.
Convert torque to power near synchronous speed.
Express rotor position relative to a synchronous reference to obtain the swing equation.
Symbols & assumptions
δ is electrical rotor angle; per-unit inertia scaling determines M. Generator, excitation and network dynamics may add states.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Use two fixed voltage magnitudes connected by a purely reactive line; for a generator use the analogous fixed internal-voltage coupling.
P/(V1V2/X)=sinδ
Solution. The lossless AC circuit gives P=(V1V2/X)sinδ.
Open this section to load its graph, or use the full-size example link below.
Orange point: horizontal coordinate 0, calculated vertical coordinate 0. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 0 on the vertical axis. Values are rounded for display.
Scope. Steady electrical-power term; the swing-equation rotor transient and voltage dynamics are not solved.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Represent conduction, convection and ventilation exchanges through appropriate conductances or mass flows.
Add internal, solar and conditioning loads.
Symbols & assumptions
Lumped sensible-heat form; humidity, surface temperatures, radiation and infiltration can require separate coupled states.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. A thermal capacitance C connects through resistance R to fixed ambient temperature. Set τ = t/(RC) and y = (T−T∞)/(T0−T∞).
y(τ)=e−τ
Solution. Separate dy/dτ = −y, integrate ln(y) = −τ + C, and apply y(0) = 1.
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Orange point: horizontal coordinate 2.5, calculated vertical coordinate 0.082085. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.082085 on the vertical axis. Values are rounded for display.
Scope. One-node constant-property cooling example; multizone and multi-node networks have additional modes.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Represents system evolution with internal states, inputs and outputs.
Component / systemFramework
Mathematical model & short derivation
Representative formulation
x˙=Ax+Buy=Cx+Du
Derivation / construction sketch
Choose independent internal variables x that determine future evolution.
Write first-order dynamics and output relations.
Linearize around an operating point to obtain the displayed linear state-space matrices.
Symbols & assumptions
A nonlinear state-space model instead uses ẋ=f(x,u), y=g(x,u); matrix dimensions and operating-point offsets must be consistent.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Use the scalar state equation y′+y=1 with y(0)=0, or transfer function 1/(s+1).
y(τ)=1−e−τ
Solution. The homogeneous response is Ce^−τ and the constant particular response is one. The initial state gives C=−1.
Open this section to load its graph, or use the full-size example link below.
Orange point: horizontal coordinate 2.5, calculated vertical coordinate 0.91792. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.91792 on the vertical axis. Values are rounded for display.
Scope. Exact linear plant reference. For bond graphs/electrical analogs use a single storage-and-resistance element; for HIL this is a reference trajectory, not measured hardware data.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Relates linear time-invariant input and output in the transform domain.
Component / systemFramework
Mathematical model & short derivation
Representative formulation
G(s)=C(sI−A)−1B+DY(s)=G(s)U(s)
Derivation / construction sketch
Take the Laplace transform of a linear time-invariant state model with zero initial conditions.
Solve (sI−A)X=BU.
Substitute into the output equation to obtain the transfer function.
Symbols & assumptions
Nonzero initial states produce additional output terms; delays and unstable dynamics require careful interpretation.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Use the scalar state equation y′+y=1 with y(0)=0, or transfer function 1/(s+1).
y(τ)=1−e−τ
Solution. The homogeneous response is Ce^−τ and the constant particular response is one. The initial state gives C=−1.
Open this section to load its graph, or use the full-size example link below.
Orange point: horizontal coordinate 2.5, calculated vertical coordinate 0.91792. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.91792 on the vertical axis. Values are rounded for display.
Scope. Exact linear plant reference. For bond graphs/electrical analogs use a single storage-and-resistance element; for HIL this is a reference trajectory, not measured hardware data.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Combines continuous dynamics with discrete state changes.
Component / systemFramework
Mathematical model & short derivation
Representative formulation
x˙=fq(x,u)q+=g(q,x,u)x+=Rq(x)
Derivation / construction sketch
Use a discrete mode q to select a continuous dynamical law.
Define guard conditions that trigger mode transitions.
Apply any state reset R at a transition.
Symbols & assumptions
Representative hybrid-system structure; event priorities and behavior at simultaneous or rapidly repeating switches must be specified.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Drop a ball from 1 m with g=9.81 m/s². At first ground contact reverse velocity and multiply its magnitude by restitution e=0.8. Plot before the second impact.
Solution. The first impact occurs at ti=√(2/g). Integrate constant gravity before and after the velocity reset with continuous height.
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Orange point: horizontal coordinate 0.55, calculated vertical coordinate 0.30139. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.55, into the displayed formula to obtain 0.30139 on the vertical axis. Values are rounded for display.
Scope. Ideal instantaneous first bounce; air resistance and contact deformation are excluded.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Represents energy exchange across mechanical, electrical and other domains.
Component / systemFramework
Mathematical model & short derivation
Representative formulation
P=ef∑flows=0at a common-effort junction
Derivation / construction sketch
Describe energy exchange using conjugate effort and flow variables.
Impose conservation of flow at equal-effort junctions and conservation of effort at equal-flow junctions.
Connect storage, dissipation and transformation elements to obtain system equations.
Symbols & assumptions
Examples: voltage/current or force/velocity. Bond orientation fixes signs; constitutive laws define each element.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Use the scalar state equation y′+y=1 with y(0)=0, or transfer function 1/(s+1).
y(τ)=1−e−τ
Solution. The homogeneous response is Ce^−τ and the constant particular response is one. The initial state gives C=−1.
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Orange point: horizontal coordinate 2.5, calculated vertical coordinate 0.91792. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.91792 on the vertical axis. Values are rounded for display.
Scope. Exact linear plant reference. For bond graphs/electrical analogs use a single storage-and-resistance element; for HIL this is a reference trajectory, not measured hardware data.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Represents accumulated quantities and their rates of change.
Component / systemFramework
Mathematical model & short derivation
Representative formulation
x˙=Fin−Fout
Derivation / construction sketch
Define x as an accumulated stock.
Apply conservation over a small time interval.
Take the interval to zero, then express flows as functions of stocks, controls and delays.
Symbols & assumptions
A model needs explicit flow laws and units; causal diagrams alone do not determine numerical predictions.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Example 1: Stock with constant inflow and linear outflow
Stock with constant inflow and linear outflow
Problem & parameters. An initially empty stock receives constant inflow q and drains at rate kS.
S/(q/k)=1−e−kt
Solution. Solve S′=q−kS with S(0)=0.
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Orange point: horizontal coordinate 2.5, calculated vertical coordinate 0.91792. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.91792 on the vertical axis. Values are rounded for display.
Scope. Single stock, constant coefficients, and no delays or saturation.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Assume the system state changes at discrete events.
Schedule candidate event times from service, arrival or failure processes.
Advance to the earliest event and update state and future schedules.
Symbols & assumptions
Discrete-event simulation is a computational framework; stochastic distributions and event rules encode the application model.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Identical events occur at Δt,2Δt,… with zero events completed at t=0.
N(t)=⌊t/Δt⌋
Solution. Count the positive integer multiples of Δt not exceeding t. The floor function gives the exact event count.
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Orange point: horizontal coordinate 3, calculated vertical coordinate 3. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 3, into the displayed formula to obtain 3 on the vertical axis. Values are rounded for display.
Scope. Simple scheduled-event benchmark; a discrete-event model need not have periodic arrivals.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Represents interacting entities following local rules.
Component / systemFramework
Mathematical model & short derivation
Representative formulation
xi(t+Δt)=Fi[xi(t),neighbors,environment,ξi]
Derivation / construction sketch
Give each agent an internal state and an interaction rule.
Compute local observations and random inputs ξi if needed.
Update states using a specified synchronous or asynchronous schedule.
Symbols & assumptions
There is no universal agent equation; local rules, spatial constraints and calibration determine emergent behavior.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Example 1: Mean position of independent moving agents
Mean position of independent moving agents
Problem & parameters. Agents start at mean position zero, have constant mean velocity 1 m/s, and do not interact.
⟨x(t)⟩=x0+vt
Solution. Each agent has x=x0+vt. Average this relation over agents; the mean is linear in time.
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Orange point: horizontal coordinate 2.5, calculated vertical coordinate 2.5. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 2.5 on the vertical axis. Values are rounded for display.
Scope. Noninteracting kinematic benchmark; not an emergent many-agent simulation.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Represents probabilistic transitions between a finite set of states.
Component / systemFramework
Mathematical model & short derivation
Representative formulation
p(t+τ)=p(t)T(τ)Tij=P[Xt+τ=j∣Xt=i]
Derivation / construction sketch
Discretize the system into states.
Estimate conditional transition probabilities at lag τ.
Assume the present state captures the relevant memory so distributions propagate by matrix multiplication.
Symbols & assumptions
Row-vector probability convention; transition rows sum to one. Molecular Markov models require lag-time and state-partition validation.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Two states exchange population at equal rate k. Initially all probability is in state one.
P1(t)=21(1+e−2kt)
Solution. Use P2=1−P1 in P1′=−kP1+kP2. Solve the resulting first-order equation.
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Orange point: horizontal coordinate 2, calculated vertical coordinate 0.50916. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 2, into the displayed formula to obtain 0.50916 on the vertical axis. Values are rounded for display.
Scope. Analytical solution or constitutive evaluation for the stated special case. It is not a general solution of the full model family.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Combines a dynamical model with noisy observations using covariance updates.
Component / systemFramework
Mathematical model & short derivation
Representative formulation
x^−=Ax^+BuK=P−HT(HP−HT+R)−1x^=x^−+K(y−Hx^−)
Derivation / construction sketch
Predict state and covariance through a linear dynamical model.
Combine predicted and measurement uncertainties.
Choose the gain minimizing posterior error covariance and update with the measurement residual.
Symbols & assumptions
Discrete linear-Gaussian Kalman filter; process covariance Q enters P⁻=APAᵀ+Q. Correlated or nonlinear errors need extensions.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Practical applications & products
Application area
Control, estimation & systems
Practical use
Estimating position from sensors under linear-Gaussian assumptions.
Problem & parameters. For one scalar measurement with observation coefficient one, hold the positive prior variance fixed.
K=P−+RP−=1+R/P−1
Solution. Insert H=1 into K=P−H/(H²P−+R).
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Orange point: horizontal coordinate 5, calculated vertical coordinate 0.16667. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 5, into the displayed formula to obtain 0.16667 on the vertical axis. Values are rounded for display.
Scope. Single measurement update; not a full dynamic filter trajectory.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Optimizes future actions using a predictive model and constraints.
Component / systemFramework
Mathematical model & short derivation
Representative formulation
mink=0∑N−1ℓ(xk,uk)+Vf(xN)xk+1=f(xk,uk)
Derivation / construction sketch
Predict a finite sequence of future states from candidate controls.
Minimize tracking or economic cost under state and input constraints.
Apply only the first control and repeat when a new state estimate arrives.
Symbols & assumptions
Model predictive control framework; stability, feasibility and execution time depend on the horizon, cost and constraints.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Practical applications & products
Application area
Control, estimation & systems
Practical use
Temperature control of a process with input limits.
Example 1: One-step unconstrained predictive control
One-step unconstrained predictive control
Problem & parameters. Let xnext=x+u and minimize (x+u)²+ρu² with ρ=1 and no constraints.
u∗=−1+ρx,ρ=1
Solution. Differentiate the quadratic cost with respect to u, set 2(x+u)+2ρu=0, and solve.
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Orange point: horizontal coordinate 0, calculated vertical coordinate -0. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain -0 on the vertical axis. Values are rounded for display.
Scope. Analytical horizon-one MPC example; longer horizons and constraints change the feedback law.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Treat the membrane as a capacitor with parallel ionic conductances.
Use voltage-dependent gate probabilities m,h,n to determine open-channel fractions.
Apply current conservation and evolve each gate with ẋ=αx(V)(1−x)−βx(V)x.
Symbols & assumptions
Classic Hodgkin–Huxley structure; conductance, reversal-potential and gating parameters are preparation-specific.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Set sodium and potassium conductances to zero, hold leak reversal potential EL fixed, and normalize V−EL by its initial value. Use τ=gLt/Cm.
y(τ)=e−τ
Solution. Separate dy/dτ = −y, integrate ln(y) = −τ + C, and apply y(0) = 1.
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Orange point: horizontal coordinate 2.5, calculated vertical coordinate 0.082085. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.082085 on the vertical axis. Values are rounded for display.
Scope. Passive leak-only reduction of Hodgkin–Huxley; action potentials and voltage-dependent gates are deliberately excluded.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Simplifies excitation and recovery into two dynamical variables.
Cross-scalePhysical model
Mathematical model & short derivation
Representative formulation
v˙=v−v3/3−w+Iw˙=ε(v+a−bw)
Derivation / construction sketch
Reduce an excitable system to a fast activation variable and slow recovery variable.
Use a cubic activation nullcline to permit threshold-like excursions.
Couple recovery to activation to produce excitation and relaxation.
Symbols & assumptions
One dimensionless FitzHugh–Nagumo convention; parameters and signs vary among formulations.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. For v′=v−v³/3−w+I set I=0 and find the zero-fast-derivative curve.
w=v−v3/3,I=0
Solution. Set v′=0 and solve for w.
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Orange point: horizontal coordinate 0, calculated vertical coordinate 0. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 0 on the vertical axis. Values are rounded for display.
Scope. A phase-plane nullcline, not a trajectory or the complete system equilibrium; equilibria also lie on the recovery nullcline.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Represents muscle mechanics with active and passive elements.
Cross-scalePhysical model
Mathematical model & short derivation
Representative formulation
(F+a)(v+b)=(F0+a)b
Derivation / construction sketch
Measure muscle force against shortening velocity under controlled activation.
Fit Hill’s hyperbolic force-velocity relation.
Combine it with force-length and passive elastic components in a practical muscle model.
Symbols & assumptions
F is force and v shortening speed; this phenomenological law is not a universal molecular derivation.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Use (F+a)(v+b)=(F0+a)b, with a/F0=0.25 and vmax=bF0/a.
F/F0=0.25+v/vmax0.25(1−v/vmax)
Solution. Solve the hyperbolic force–velocity equation for F and substitute the normalized speed.
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Orange point: horizontal coordinate 0.5, calculated vertical coordinate 0.16667. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 0.16667 on the vertical axis. Values are rounded for display.
Scope. Steady concentric shortening only; activation and length effects are held fixed.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Represents vascular resistance and compliance with lumped elements.
Cross-scalePhysical model
Mathematical model & short derivation
Representative formulation
CdtdP=Qin−RP−Pv
Derivation / construction sketch
Represent arterial storage by compliance C and peripheral outflow by resistance R.
Use stored-volume change dV=C dP.
Apply flow conservation to obtain the pressure equation.
Symbols & assumptions
Two-element Windkessel; Pv is downstream pressure. Three- and four-element variants improve characteristic impedance or inertia representation.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. With zero inflow, a two-element Windkessel discharges through resistance R from compliance C. Use τ=t/(RC) and normalize pressure above venous pressure.
y(τ)=e−τ
Solution. Separate dy/dτ = −y, integrate ln(y) = −τ + C, and apply y(0) = 1.
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Orange point: horizontal coordinate 2.5, calculated vertical coordinate 0.082085. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.082085 on the vertical axis. Values are rounded for display.
Scope. Constant-compliance diastolic interval, not a full pulsatile cardiac cycle.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Adds perfusion and metabolic heat to tissue heat transfer.
Cross-scalePhysical model
Mathematical model & short derivation
Representative formulation
ρc∂tT=∇⋅(k∇T)+ρbcbωb(Ta−T)+Qmet+Qext
Derivation / construction sketch
Start with tissue heat storage and conduction.
Approximate perfusion exchange by blood entering at arterial temperature Ta and equilibrating locally.
Add metabolic and applied heating.
Symbols & assumptions
Pennes model; ωb is volumetric perfusion per tissue volume. Large-vessel directional heat transfer is not resolved.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Take spatially uniform tissue with constant heat source Q, blood heat-exchange coefficient W>0, and initial tissue temperature equal to arterial temperature Ta.
(T−Ta)/(Q/W)=1−e−Wt/(ρc)
Solution. The Pennes balance reduces to ρc T′=Q−W(T−Ta). Solve the linear initial-value problem.
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Orange point: horizontal coordinate 2.5, calculated vertical coordinate 0.91792. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.91792 on the vertical axis. Values are rounded for display.
Scope. Uniform-temperature reduction; no spatial conduction, temperature-dependent perfusion, or safety prediction.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Couple local reaction kinetics to diffusion of two species.
Linearize about a homogeneous steady state.
Compare eigenvalues with and without diffusion to identify diffusion-driven pattern instability.
Symbols & assumptions
Turing-type reaction-diffusion family; a pattern requires suitable kinetics and diffusion contrast, not merely the presence of diffusion.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Solve ut=uxx−u with zero ends and initial sin(πx), then plot t=1.
u(x,1)=e−(π2+1)sin(πx)
Solution. The Laplacian and decay each multiply the mode by a negative constant; its amplitude solves A′=−(π²+1)A.
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Orange point: horizontal coordinate 0.5, calculated vertical coordinate 1.9028e-05. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 1.9028e-05 on the vertical axis. Values are rounded for display.
Scope. One-species linear stable subproblem, not a two-species Turing pattern or nonlinear morphogenesis prediction.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Fit growth rate to a saturating function of limiting substrate S.
At low substrate it is approximately linear; at high substrate it approaches μmax.
Multiply specific growth by biomass X and include decay if appropriate.
Symbols & assumptions
Monod model; KS is half-saturation concentration and kd decay rate. Substrate mass balance is also required.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Evaluate growth rate at prescribed substrate concentration with fixed Monod parameters.
μ/μmax=S/(Ks+S)
Solution. Normalize μ=μmax S/(Ks+S) by μmax and substitute S/Ks.
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Orange point: horizontal coordinate 4, calculated vertical coordinate 0.8. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 4, into the displayed formula to obtain 0.8 on the vertical axis. Values are rounded for display.
Scope. Growth-rate relation only; substrate depletion and biomass evolution are not integrated.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Represents exchange between anatomically motivated compartments.
Cross-scalePhysical model
Mathematical model & short derivation
Representative formulation
VidtdCi=Qi(Ca−Ci/Ki)−CLiCi
Derivation / construction sketch
Represent an organ as a well-mixed compartment.
Balance arterial delivery against venous removal using a partition relation.
Subtract local clearance or transformation when applicable.
Symbols & assumptions
Representative perfusion-limited PBPK compartment; Vi is volume, Qi perfusion, Ki tissue-blood partition coefficient and CLi a compatible clearance term.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. After an initial dose, use one well-mixed compartment with first-order elimination, no further input, and τ=kt.
y(τ)=e−τ
Solution. Separate dy/dτ = −y, integrate ln(y) = −τ + C, and apply y(0) = 1.
Open this section to load its graph, or use the full-size example link below.
Orange point: horizontal coordinate 2.5, calculated vertical coordinate 0.082085. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.082085 on the vertical axis. Values are rounded for display.
Scope. One-compartment limiting case; interorgan exchange, binding, and nonlinear metabolism are excluded.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Example 1: Homogeneous Maxwellian velocity marginal
Homogeneous Maxwellian velocity marginal
Problem & parameters. Take a spatially uniform equilibrium with zero drift and the normalized Gaussian velocity marginal. For collisionless plasma use zero fields and a neutralizing background.
vthf(v)=π−1/2e−(v/vth)2
Solution. The homogeneous force-free streaming terms vanish. Maxwellian collisions balance for Boltzmann equilibrium; integrating the Gaussian fixes its normalization.
Open this section to load its graph, or use the full-size example link below.
Orange point: horizontal coordinate 0, calculated vertical coordinate 0.56419. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 0.56419 on the vertical axis. Values are rounded for display.
Scope. Equilibrium distribution or exact kinetic benchmark. DSMC and PIC would estimate it using particles; this plot is not a finite-particle sample.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Couples collisionless distribution dynamics to electrostatic fields.
Cross-scalePhysical model
Mathematical model & short derivation
Representative formulation
∂tf+v⋅∇xf−mq∇ϕ⋅∇vf=0−ε0∇2ϕ=ρ
Derivation / construction sketch
Neglect collisions in the kinetic transport equation.
Restrict fields to electrostatics with E=−∇φ.
Integrate each species distribution over velocity to obtain charge density for Poisson’s equation.
Symbols & assumptions
Multiple species contribute ρ=Σs qs∫fsdv; electrostatic approximation omits electromagnetic induction and radiation.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Example 1: Homogeneous Maxwellian velocity marginal
Homogeneous Maxwellian velocity marginal
Problem & parameters. Take a spatially uniform equilibrium with zero drift and the normalized Gaussian velocity marginal. For collisionless plasma use zero fields and a neutralizing background.
vthf(v)=π−1/2e−(v/vth)2
Solution. The homogeneous force-free streaming terms vanish. Maxwellian collisions balance for Boltzmann equilibrium; integrating the Gaussian fixes its normalization.
Open this section to load its graph, or use the full-size example link below.
Orange point: horizontal coordinate 0, calculated vertical coordinate 0.56419. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 0.56419 on the vertical axis. Values are rounded for display.
Scope. Equilibrium distribution or exact kinetic benchmark. DSMC and PIC would estimate it using particles; this plot is not a finite-particle sample.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Couples collisionless kinetic distributions to electromagnetic fields.
Cross-scalePhysical model
Mathematical model & short derivation
Representative formulation
∂tfs+v⋅∇xfs+msqs(E+v×B)⋅∇vfs=0
Derivation / construction sketch
Neglect collisional scattering while retaining the Lorentz force.
Compute charge and current by taking velocity moments of all species distributions.
Use those moments as sources in Maxwell’s equations.
Symbols & assumptions
Nonrelativistic phase-space form shown; relativistic momentum coordinates are required for high-energy particles.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Example 1: Homogeneous Maxwellian velocity marginal
Homogeneous Maxwellian velocity marginal
Problem & parameters. Take a spatially uniform equilibrium with zero drift and the normalized Gaussian velocity marginal. For collisionless plasma use zero fields and a neutralizing background.
vthf(v)=π−1/2e−(v/vth)2
Solution. The homogeneous force-free streaming terms vanish. Maxwellian collisions balance for Boltzmann equilibrium; integrating the Gaussian fixes its normalization.
Open this section to load its graph, or use the full-size example link below.
Orange point: horizontal coordinate 0, calculated vertical coordinate 0.56419. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 0.56419 on the vertical axis. Values are rounded for display.
Scope. Equilibrium distribution or exact kinetic benchmark. DSMC and PIC would estimate it using particles; this plot is not a finite-particle sample.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Treats a conducting fluid coupled to a magnetic field.
Cross-scalePhysical model
Mathematical model & short derivation
Representative formulation
ρDtDu=−∇p+J×B∂tB=∇×(u×B)+ηm∇2B
Derivation / construction sketch
Take velocity moments of kinetic species equations and form a conducting-fluid description.
Use a resistive Ohm law and Ampère’s law without displacement current.
Combine with Faraday induction to obtain magnetic-field evolution.
Symbols & assumptions
Simple resistive MHD with constant magnetic diffusivity ηm; continuity, energy and ∇·B=0 complete the system.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Use a one-dimensional sinusoidal wave in a uniform, lossless linear medium. Plot the normalized field at time zero.
u(ξ,0)=sin(2πξ)
Solution. A sinusoidal traveling-wave solution is u = sin[2π(ξ−τ)]. Set τ = 0 to obtain the plotted snapshot.
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Orange point: horizontal coordinate 0.5, calculated vertical coordinate 1.2246e-16. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 1.2246e-16 on the vertical axis. Values are rounded for display.
Scope. An acoustic, electromagnetic, elastic, or linear Alfvén-wave reference as appropriate. For MHD this is the small transverse perturbation of a uniform magnetized equilibrium; for FDTD it is an exact target, not a discretized result.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Tracks neutron angular and energy-dependent transport with interactions.
Cross-scalePhysical model
Mathematical model & short derivation
Representative formulation
v1∂tψ+Ω⋅∇ψ+Σtψ=∫Σsψ′dΩ′dE′+Sf+Q
Derivation / construction sketch
Balance angular neutron flux in a spatial, directional and energy element.
Subtract streaming losses and collision removal.
Add scattering into the element, fission emission and external sources.
Symbols & assumptions
ψ is angular flux, Σ cross sections and Sf fission source. Energy dependence and boundary conditions are essential.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. A steady beam traverses a homogeneous purely absorbing medium. Set τ = Σx and y = intensity / incident intensity.
y(τ)=e−τ
Solution. Separate dy/dτ = −y, integrate ln(y) = −τ + C, and apply y(0) = 1.
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Orange point: horizontal coordinate 2.5, calculated vertical coordinate 0.082085. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.082085 on the vertical axis. Values are rounded for display.
Scope. Exact absorption-only transport benchmark, without scattering or emission. For Monte Carlo transport this is the expected value, not a sampled realization.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
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Simplifies neutron transport to a diffusion description.
Cross-scalePhysical model
Mathematical model & short derivation
Representative formulation
v1∂tϕ−∇⋅(D∇ϕ)+Σaϕ=S
Derivation / construction sketch
Integrate neutron transport over directions to obtain a scalar-flux balance.
Approximate angular distribution as nearly isotropic.
Close current by Fick-like leakage J=−D∇φ.
Symbols & assumptions
Diffusion coefficient is often D≈1/(3Σtr); interfaces, voids and strongly absorbing boundaries challenge the approximation.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Practical applications & products
Application area
Plasma, nuclear & astrophysics
Practical use
Flux distribution where angular anisotropy is weak.
Example 1: Subcritical neutron-density diffusion mode
Subcritical neutron-density diffusion mode
Problem & parameters. On a slab solve nτ=nξξ with zero extrapolated-end values and initial sin(πξ), ignoring reactions in this illustrative diffusion subproblem.
u(ξ,τ)=sin(πξ)e−π2τ,τ=0.1
Solution. The sine satisfies both zero end values. Its second derivative is −π² times itself; the amplitude solves a′ = −π²a.
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Orange point: horizontal coordinate 0.5, calculated vertical coordinate 0.37271. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 0.37271 on the vertical axis. Values are rounded for display.
Scope. Diffusion-only benchmark; absorption and fission terms would modify the mode growth/decay rate.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Approximates time-dependent neutron population with delayed-neutron groups.
Cross-scalePhysical model
Mathematical model & short derivation
Representative formulation
n˙=Λρreact−βn+i∑λiCiC˙i=Λβin−λiCi
Derivation / construction sketch
Assume the spatial neutron-flux shape is fixed while its amplitude changes.
Separate prompt neutrons from delayed-neutron precursor groups.
Balance neutron population and precursor production/decay.
Symbols & assumptions
n is neutron amplitude, Λ generation time, β delayed fraction and ρreact reactivity; thermal feedback may be coupled separately.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Set delayed-neutron fraction and external source to zero, take constant negative reactivity ρ, and prompt generation time Λ.
n/n0=e−τ,τ=∣ρ∣t/Λ
Solution. Point kinetics reduces to n′=(ρ/Λ)n. Integrate with n(0)=n0.
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Orange point: horizontal coordinate 2.5, calculated vertical coordinate 0.082085. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.082085 on the vertical axis. Values are rounded for display.
Scope. Prompt-only idealization, not a realistic startup, shutdown, or reactor-safety calculation.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
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Evolves coupled radioactive parent and daughter populations.
Cross-scalePhysical model
Mathematical model & short derivation
Representative formulation
N˙i=j∑bj→iλjNj−λiNi
Derivation / construction sketch
Treat each unstable nuclide as having an exponential decay probability per time.
Add production from parent decays and subtract its own decay.
Collect the coupled linear system and solve with a matrix exponential or chain formula.
Symbols & assumptions
λ are decay constants and b branching fractions. Irradiation adds reaction production/removal terms.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Example 1: Daughter buildup in a two-step decay chain
Daughter buildup in a two-step decay chain
Problem & parameters. Initially N1=N10 and N2=0. Let the parent decay to the daughter with λ2=2λ1 and unit branching fraction.
N2/N10=e−λ1t−e−2λ1t
Solution. Solve N1=N10exp(−λ1t). Insert into N2′+λ2N2=λ1N1 and integrate using an integrating factor.
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Orange point: horizontal coordinate 2.5, calculated vertical coordinate 0.075347. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.075347 on the vertical axis. Values are rounded for display.
Scope. Analytical solution or constitutive evaluation for the stated special case. It is not a general solution of the full model family.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
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Evolves masses under mutual inverse-square attraction.
Cross-scalePhysical model
Mathematical model & short derivation
Representative formulation
r¨i=−Gj=i∑∣ri−rj∣3mj(ri−rj)
Derivation / construction sketch
Apply Newton’s inverse-square gravitational force to every pair of masses.
Sum all forces on a selected body.
Divide by that body’s mass and integrate the coupled trajectories.
Symbols & assumptions
Point masses under nonrelativistic gravity; close encounters, collisions and extended bodies may need special treatment.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Reduce an isolated gravitational system to two point masses with total mass M in a circular relative orbit of radius a.
x/a=cos(nt),n=GM/a3
Solution. Balance relative centripetal acceleration n²a against GM/a². The Cartesian x coordinate then follows a cosine.
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Orange point: horizontal coordinate 3.1416, calculated vertical coordinate -1. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 3.1416, into the displayed formula to obtain -1 on the vertical axis. Values are rounded for display.
Scope. Exact two-body circular orbit, not a general many-body solution; a one-coordinate time trace is shown.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Describe gravitation through a spacetime metric rather than a Newtonian force field.
Vary the Einstein–Hilbert action plus matter action with respect to the metric.
Stationarity gives the Einstein field equations.
Symbols & assumptions
Gμν is the Einstein curvature tensor, Tμν stress-energy and Λ cosmological constant. This is a relativistic field theory; coordinate and boundary choices matter.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Example 1: Gravitational time dilation outside a sphere
Gravitational time dilation outside a sphere
Problem & parameters. For a stationary observer outside a nonrotating spherical mass, compare proper time with Schwarzschild coordinate time at infinity.
dτ/dt=1−rs/r
Solution. Set spatial coordinate increments to zero in the Schwarzschild line element and take the square root of its time coefficient.
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Orange point: horizontal coordinate 4.525, calculated vertical coordinate 0.88261. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 4.525, into the displayed formula to obtain 0.88261 on the vertical axis. Values are rounded for display.
Scope. Exterior vacuum Schwarzschild solution, r>rs. A static observer cannot remain at the horizon.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
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Couples hydrostatic balance, energy transport and energy generation.
Cross-scalePhysical model
Mathematical model & short derivation
Representative formulation
drdm=4πr2ρdrdP=−r2GmρdrdL=4πr2ρε
Derivation / construction sketch
Apply spherical mass conservation.
Balance gravity with the pressure gradient for hydrostatic support.
Integrate local energy generation into luminosity and add an energy-transport law.
Symbols & assumptions
Quasi-static spherical stellar structure; an equation of state, opacity and nuclear reaction network close the model.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Use an ideal gas at constant temperature and constant gravity, with density ρ0 at height zero.
ρ(z)/ρ0=e−z/H
Solution. Combine dp/dz=−ρg with p=ρRsT; integrate dρ/dz=−ρ/H, H=RsT/g.
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Orange point: horizontal coordinate 2.5, calculated vertical coordinate 0.082085. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.082085 on the vertical axis. Values are rounded for display.
Scope. Hydrostatic column benchmark only. For stellar structure this approximates a thin isothermal layer, not an entire star; radiation, convection, and dynamics are excluded.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
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Assumes a homogeneous and isotropic expanding spacetime.
Cross-scalePhysical model
Mathematical model & short derivation
Representative formulation
H2=38πGρ−a2kc2+3Λc2H=aa˙
Derivation / construction sketch
Assume large-scale spatial homogeneity and isotropy, defining an FLRW metric.
Insert it into Einstein’s equations.
The time-time component gives the first Friedmann equation for the scale factor a.
Symbols & assumptions
ρ is mass-equivalent energy density; a pressure relation and conservation equation determine evolution. k is spatial curvature in a consistent normalization.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Take a spatially flat FLRW universe with pressureless matter only and zero cosmological constant.
a(t)/a(t∗)=(t/t∗)2/3
Solution. Mass conservation gives ρ∝a^−3. Friedmann’s equation then gives ȧ∝a^−1/2; integrate from the big-bang branch.
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Orange point: horizontal coordinate 1.51, calculated vertical coordinate 1.3162. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 1.51, into the displayed formula to obtain 1.3162 on the vertical axis. Values are rounded for display.
Scope. Matter-only special case, not a fit to the present universe.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Approximates fields with basis functions over elements.
Cross-scaleNumerical method
Mathematical model & short derivation
Representative formulation
Ku=fKij=∫Ω∇Ni⋅k∇NjdΩ
Derivation / construction sketch
For a representative diffusion equation, multiply by a test function and integrate by parts.
Approximate the field by basis functions Ni and choose matching test functions.
Assemble the resulting element integrals into a global matrix system.
Symbols & assumptions
Displayed weak form is for scalar diffusion; FEM itself is not a constitutive model. Essential and natural boundary conditions must be treated consistently.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Practical applications & products
Application area
Numerical solution methods
Practical use
Solving a structural elasticity boundary-value problem.
Example 1: Steady one-dimensional diffusion benchmark
Steady one-dimensional diffusion benchmark
Problem & parameters. Solve u″ = 0 on 0 < ξ < 1 with u(0) = 1 and u(1) = 0, constant transport coefficient, and no source.
u(ξ)=1−ξ
Solution. Integrate twice to obtain u = A+Bξ. The two endpoint values give A = 1 and B = −1.
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Orange point: horizontal coordinate 0.5, calculated vertical coordinate 0.5. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 0.5 on the vertical axis. Values are rounded for display.
Scope. For numerical-method entries this is the exact target to verify against, not a computed discretization or convergence claim.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Discretizes conservation laws using fluxes across control-volume boundaries.
Cross-scaleNumerical method
Mathematical model & short derivation
Representative formulation
VidtdUi+faces∑Ff⋅nfAf=ViSi
Derivation / construction sketch
Integrate a conservation law over a control volume.
Use the divergence theorem to convert volume flux divergence into surface fluxes.
Approximate each face flux while sharing it consistently between adjacent cells.
Symbols & assumptions
U is a conserved quantity, F flux and S source; numerical reconstruction and flux choices determine accuracy and stability.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Example 1: Steady one-dimensional diffusion benchmark
Steady one-dimensional diffusion benchmark
Problem & parameters. Solve u″ = 0 on 0 < ξ < 1 with u(0) = 1 and u(1) = 0, constant transport coefficient, and no source.
u(ξ)=1−ξ
Solution. Integrate twice to obtain u = A+Bξ. The two endpoint values give A = 1 and B = −1.
Open this section to load its graph, or use the full-size example link below.
Orange point: horizontal coordinate 0.5, calculated vertical coordinate 0.5. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 0.5 on the vertical axis. Values are rounded for display.
Scope. For numerical-method entries this is the exact target to verify against, not a computed discretization or convergence claim.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Approximates derivatives with differences on a grid.
Cross-scaleNumerical method
Mathematical model & short derivation
Representative formulation
∂x2∂2u≈Δx2ui+1−2ui+ui−1
Derivation / construction sketch
Expand neighboring function values in Taylor series about grid point i.
Add the two expansions so odd derivatives cancel.
Solve for the second derivative, leaving an O(Δx²) truncation error.
Symbols & assumptions
Uniform grid and sufficiently smooth u; boundary formulas and time stepping determine the full discretization.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Example 1: Steady one-dimensional diffusion benchmark
Steady one-dimensional diffusion benchmark
Problem & parameters. Solve u″ = 0 on 0 < ξ < 1 with u(0) = 1 and u(1) = 0, constant transport coefficient, and no source.
u(ξ)=1−ξ
Solution. Integrate twice to obtain u = A+Bξ. The two endpoint values give A = 1 and B = −1.
Open this section to load its graph, or use the full-size example link below.
Orange point: horizontal coordinate 0.5, calculated vertical coordinate 0.5. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 0.5 on the vertical axis. Values are rounded for display.
Scope. For numerical-method entries this is the exact target to verify against, not a computed discretization or convergence claim.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Recasts suitable field problems as boundary integral equations.
Cross-scaleNumerical method
Mathematical model & short derivation
Representative formulation
c(P)u(P)+∫Γu∂nGdΓ=∫ΓG∂nudΓ
Derivation / construction sketch
Choose a fundamental solution G of the governing linear differential operator.
Apply Green’s identity to the field and G.
Move the problem to the boundary and discretize the boundary unknowns.
Symbols & assumptions
Representative Laplace boundary-integral equation; c(P) depends on geometry and limiting convention. Nonlinear or heterogeneous media need extensions.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Example 1: Steady one-dimensional diffusion benchmark
Steady one-dimensional diffusion benchmark
Problem & parameters. Solve u″ = 0 on 0 < ξ < 1 with u(0) = 1 and u(1) = 0, constant transport coefficient, and no source.
u(ξ)=1−ξ
Solution. Integrate twice to obtain u = A+Bξ. The two endpoint values give A = 1 and B = −1.
Open this section to load its graph, or use the full-size example link below.
Orange point: horizontal coordinate 0.5, calculated vertical coordinate 0.5. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 0.5 on the vertical axis. Values are rounded for display.
Scope. For numerical-method entries this is the exact target to verify against, not a computed discretization or convergence claim.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Represents fields with global or element-wise high-order basis expansions.
Cross-scaleNumerical method
Mathematical model & short derivation
Representative formulation
uN(x)=n=0∑Nanϕn(x)
Derivation / construction sketch
Expand the solution in a global or element-local basis.
Insert the expansion into the governing equation.
Set weighted residuals or collocation residuals to zero to solve for coefficients an.
Symbols & assumptions
Fourier, Chebyshev and other bases suit different domains; smoothness drives convergence and discontinuities can cause oscillations.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Use uτ = uξξ on the unit interval, zero end values, and u(ξ,0) = sin(πξ). Plot τ = 0.1.
u(ξ,τ)=sin(πξ)e−π2τ,τ=0.1
Solution. The sine satisfies both zero end values. Its second derivative is −π² times itself; the amplitude solves a′ = −π²a.
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Orange point: horizontal coordinate 0.5, calculated vertical coordinate 0.37271. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 0.37271 on the vertical axis. Values are rounded for display.
Scope. Exact PDE benchmark. For reduced bases, PINNs, and neural operators, this is a reference target, not a claimed trained or computed prediction.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
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Approximates continuum fields through moving particles and kernels.
Cross-scaleNumerical method
Mathematical model & short derivation
Representative formulation
f(ri)≈j∑mjρjfjW(ri−rj,h)
Derivation / construction sketch
Approximate a field by convolution with a smoothing kernel.
Replace the volume integral with particle volumes mj/ρj.
Differentiate the kernel to construct discrete gradients and conservation equations.
Symbols & assumptions
W is a normalized kernel with smoothing length h; boundary consistency, tensile instability and conservation depend on the chosen formulation.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Evaluate the standard one-dimensional cubic-spline smoothing kernel of support radius 2h.
hW(q)=32{1−1.5q2+0.75q3(2−q)3/4q<11≤q≤2
Solution. Use q=|x|/h, apply the inner and outer polynomial branches, and normalize their integral to one.
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Orange point: horizontal coordinate 0, calculated vertical coordinate 0.66667. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 0.66667 on the vertical axis. Values are rounded for display.
Scope. Kernel evaluation, not a complete SPH flow or solid simulation.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
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Evolves contacting discrete bodies with contact laws.
Cross-scaleNumerical method
Mathematical model & short derivation
Representative formulation
mir¨i=j∑Fij+migIiω˙i=j∑Mij
Derivation / construction sketch
Treat grains as individual bodies.
Compute overlap- or geometry-based contact forces and frictional moments.
Apply translational and rotational momentum balances to each grain.
Symbols & assumptions
DEM is a numerical framework; normal stiffness, damping, friction and cohesion laws must be specified and calibrated.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Choose a linear frictionless normal-contact spring with stiffness k, no damping, and positive overlap.
F/(kδ∗)=δ/δ∗
Solution. The prescribed contact law is F=kδ. Before contact, F=0.
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Orange point: horizontal coordinate 0.5, calculated vertical coordinate 0.5. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 0.5 on the vertical axis. Values are rounded for display.
Scope. One elastic contact contribution; many-particle dynamics and tangential friction are excluded.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
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Evolves discrete velocity populations to recover suitable macroscopic flow equations.
Cross-scaleNumerical method
Mathematical model & short derivation
Representative formulation
fi(x+ciΔt,t+Δt)=fi(x,t)−τΔt[fi−fieq]
Derivation / construction sketch
Discretize particle velocity space into lattice directions ci.
Alternate streaming with relaxation toward a local equilibrium distribution.
Take a long-wavelength, low-Mach expansion to recover continuum hydrodynamics.
Symbols & assumptions
Single-relaxation-time LBM example; viscosity relates to τ and Δt. More robust collision operators and boundary schemes are common.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Use a small-amplitude periodic transverse shear wave in the low-Mach hydrodynamic limit; plot νt/L²=0.02.
u(ξ,τ)=sin(2πξ)e−4π2τ,τ=0.02
Solution. The continuum transverse velocity obeys diffusion. Its wave number 2π/L fixes the exponential decay rate.
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Orange point: horizontal coordinate 0.5, calculated vertical coordinate 5.5604e-17. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 5.5604e-17 on the vertical axis. Values are rounded for display.
Scope. Exact continuum benchmark for LBM, not a finite-lattice prediction; compressibility and lattice errors must be checked separately.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Resolves turbulence without a turbulence closure for the selected flow equations.
Cross-scaleNumerical method
Mathematical model & short derivation
Representative formulation
∂tu+u⋅∇u=−ρ∇p+ν∇2uΔx must resolve dissipative scales
Derivation / construction sketch
Choose the physical flow equations without an added turbulence closure.
Resolve the energy-containing and dissipative motions with sufficiently fine space and time steps.
Check convergence and conservation to assess numerical resolution.
Symbols & assumptions
DNS is a resolution strategy, not a new fluid law; feasible Reynolds numbers are limited by computational cost.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Practical applications & products
Application area
Numerical solution methods
Practical use
A research simulation at a computationally tractable Reynolds number.
Problem & parameters. Take steady, fully developed incompressible flow with constant viscosity between fixed parallel plates. For Hagen–Poiseuille use the equivalent diameter cut through a round pipe.
u/Umax=1−ξ2
Solution. The axial momentum equation becomes a constant second derivative. Integrate twice and impose no slip at both walls to obtain a parabola.
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Orange point: horizontal coordinate 0, calculated vertical coordinate 1. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 1 on the vertical axis. Values are rounded for display.
Scope. Exact laminar benchmark. Plate and pipe pressure-to-maximum-speed factors differ; the plotted normalized profile is identical. DNS here resolves this simple laminar case.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Samples particle motion and collisions in a rarefied gas.
Cross-scaleNumerical method
Mathematical model & short derivation
Representative formulation
Ppair∝VcellσT(g)gΔt
Derivation / construction sketch
Split rarefied-gas evolution into particle motion and collisions over a short time step.
Select representative collision pairs in local cells using relative speed g and total cross section σT.
Sample post-collision states while conserving the appropriate quantities.
Symbols & assumptions
DSMC acceptance also depends on particle statistical weights and collision-selection scheme; cell/time scales must resolve mean-free-path and collision-time behavior.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Example 1: Homogeneous Maxwellian velocity marginal
Homogeneous Maxwellian velocity marginal
Problem & parameters. Take a spatially uniform equilibrium with zero drift and the normalized Gaussian velocity marginal. For collisionless plasma use zero fields and a neutralizing background.
vthf(v)=π−1/2e−(v/vth)2
Solution. The homogeneous force-free streaming terms vanish. Maxwellian collisions balance for Boltzmann equilibrium; integrating the Gaussian fixes its normalization.
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Orange point: horizontal coordinate 0, calculated vertical coordinate 0.56419. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 0.56419 on the vertical axis. Values are rounded for display.
Scope. Equilibrium distribution or exact kinetic benchmark. DSMC and PIC would estimate it using particles; this plot is not a finite-particle sample.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Move computational particles under interpolated fields.
Deposit particle charge and current on a mesh.
Solve field equations and gather fields back to particles for the next step.
Symbols & assumptions
PIC scheme; charge-conserving deposition, field solver, particle shape W and time integration determine numerical behavior.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Example 1: Homogeneous Maxwellian velocity marginal
Homogeneous Maxwellian velocity marginal
Problem & parameters. Take a spatially uniform equilibrium with zero drift and the normalized Gaussian velocity marginal. For collisionless plasma use zero fields and a neutralizing background.
vthf(v)=π−1/2e−(v/vth)2
Solution. The homogeneous force-free streaming terms vanish. Maxwellian collisions balance for Boltzmann equilibrium; integrating the Gaussian fixes its normalization.
Open this section to load its graph, or use the full-size example link below.
Orange point: horizontal coordinate 0, calculated vertical coordinate 0.56419. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 0.56419 on the vertical axis. Values are rounded for display.
Scope. Equilibrium distribution or exact kinetic benchmark. DSMC and PIC would estimate it using particles; this plot is not a finite-particle sample.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Discretize Maxwell’s curl equations on staggered spatial locations.
Stagger electric and magnetic updates by half a time step.
Alternate the updates to propagate electromagnetic fields.
Symbols & assumptions
Yee-type FDTD; Courant stability, absorbing boundaries and dispersive material updates must be respected.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Use a one-dimensional sinusoidal wave in a uniform, lossless linear medium. Plot the normalized field at time zero.
u(ξ,0)=sin(2πξ)
Solution. A sinusoidal traveling-wave solution is u = sin[2π(ξ−τ)]. Set τ = 0 to obtain the plotted snapshot.
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Orange point: horizontal coordinate 0.5, calculated vertical coordinate 1.2246e-16. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 1.2246e-16 on the vertical axis. Values are rounded for display.
Scope. An acoustic, electromagnetic, elastic, or linear Alfvén-wave reference as appropriate. For MHD this is the small transverse perturbation of a uniform magnetized equilibrium; for FDTD it is an exact target, not a discretized result.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Transfers particle-carried material state to a computational grid.
Cross-scaleNumerical method
Mathematical model & short derivation
Representative formulation
mi=p∑mpNi(xp)fi,int=−p∑Vpσp⋅∇Ni(xp)
Derivation / construction sketch
Store mass, stress and history on material particles.
Transfer mass and internal forces to a background grid using shape functions Ni.
Solve grid momentum and transfer updated motion back to particles.
Symbols & assumptions
Representative MPM mapping; transfer choices, cell crossing and boundary conditions affect conservation and accuracy.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Apply a uniform small axial strain of 0.01 to a homogeneous elastic bar.
u(x)/L=0.01(x/L)
Solution. Integrate du/dx=0.01 with u(0)=0.
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Orange point: horizontal coordinate 0.5, calculated vertical coordinate 0.005. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 0.005 on the vertical axis. Values are rounded for display.
Scope. Exact continuum target for MPM; grid transfer and particle quadrature errors are not represented.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Samples particle histories and interactions statistically.
Cross-scaleNumerical method
Mathematical model & short derivation
Representative formulation
s=−Σtlnξtally estimate=N1k∑wkfk
Derivation / construction sketch
Sample a free path from the exponential survival law in a homogeneous material.
Sample interaction type and outgoing state using cross sections.
Average weighted particle-history contributions to estimate observables and sampling error.
Symbols & assumptions
ξ is uniform on (0,1); heterogeneous materials require boundary tracking. Variance-reduction weights must preserve unbiased tallies.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Practical applications & products
Application area
Numerical solution methods
Practical use
Radiation shielding calculations with sampling uncertainty.
Problem & parameters. A steady beam traverses a homogeneous purely absorbing medium. Set τ = Σx and y = intensity / incident intensity.
y(τ)=e−τ
Solution. Separate dy/dτ = −y, integrate ln(y) = −τ + C, and apply y(0) = 1.
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Orange point: horizontal coordinate 2.5, calculated vertical coordinate 0.082085. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.082085 on the vertical axis. Values are rounded for display.
Scope. Exact absorption-only transport benchmark, without scattering or emission. For Monte Carlo transport this is the expected value, not a sampled realization.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Derives effective properties or equations from smaller-scale structure.
Cross-scaleFramework
Mathematical model & short derivation
Representative formulation
σˉ=Ceff:εˉσˉ=⟨σ⟩εˉ=⟨ε⟩
Derivation / construction sketch
Solve a microscale boundary-value problem under imposed macroscopic loading.
Average stress and strain over a representative region.
Define an effective constitutive relation consistent with those averages.
Symbols & assumptions
Linear elastic example; scale separation, boundary conditions and statistical representativeness determine validity.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Two perfectly bonded parallel axial bars share the same strain, with modulus ratio E2/E1=4.
Eeff/E1=(1−f)+4f
Solution. Average stress is [(1−f)E1+fE2] times the common strain. Divide by strain to obtain the effective axial modulus.
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Orange point: horizontal coordinate 0.5, calculated vertical coordinate 2.5. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 2.5 on the vertical axis. Values are rounded for display.
Scope. Exact iso-strain parallel-bar construction; generally an upper-bound estimate for other microstructures.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Uses a finite microstructural sample to estimate bulk response.
Cross-scaleFramework
Mathematical model & short derivation
Representative formulation
KeffG=⟨k(x)(G+∇w)⟩
Derivation / construction sketch
Choose a sample of heterogeneous material and impose average temperature gradient G.
Solve for the microscopic fluctuation field w with compatible boundary conditions.
Average the flux to infer effective conductivity.
Symbols & assumptions
Representative volume element conductivity example; tensor columns follow from independent applied gradients. An undersized sample may not be representative.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Practical applications & products
Application area
Multiscale, reduced & data-driven models
Practical use
Effective conductivity of a heterogeneous material.
Problem & parameters. Apply uniform uniaxial strain to a homogeneous small-strain elastic bar with traction-free lateral surfaces.
σ/E=ε
Solution. The one-dimensional constitutive law is σ = Eε; divide by E. For an orthotropic solid use its modulus along a principal material axis.
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Orange point: horizontal coordinate 0.005, calculated vertical coordinate 0.005. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.005, into the displayed formula to obtain 0.005 on the vertical axis. Values are rounded for display.
Scope. Homogeneous linear reference for truss, RVE, and FE² entries; this is not a heterogeneous microscale simulation.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Solves microscale problems within a macroscale finite-element calculation.
Cross-scaleFramework
Mathematical model & short derivation
Representative formulation
σˉ(εˉ)=∣Ωmicro∣1∫Ωmicroσ(εˉ+∇su~)dV
Derivation / construction sketch
At each macroscale integration point, impose its strain on a microscale problem.
Solve the microstructure’s equilibrium with compatible fluctuation boundaries.
Return volume-averaged stress and a consistent tangent to the macroscale solver.
Symbols & assumptions
Small-strain FE² example; ∇s is the symmetric gradient. Computational cost and micro-macro energy consistency are central.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Apply uniform uniaxial strain to a homogeneous small-strain elastic bar with traction-free lateral surfaces.
σ/E=ε
Solution. The one-dimensional constitutive law is σ = Eε; divide by E. For an orthotropic solid use its modulus along a principal material axis.
Open this section to load its graph, or use the full-size example link below.
Orange point: horizontal coordinate 0.005, calculated vertical coordinate 0.005. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.005, into the displayed formula to obtain 0.005 on the vertical axis. Values are rounded for display.
Scope. Homogeneous linear reference for truss, RVE, and FE² entries; this is not a heterogeneous microscale simulation.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Combines quantum mechanics in a selected region with molecular mechanics around it.
Cross-scaleFramework
Mathematical model & short derivation
Representative formulation
Etotal=EQM+EMM+Ecoupling
Derivation / construction sketch
Partition the system into a quantum region and a molecular-mechanics environment.
Evaluate each region with its chosen representation.
Add compatible electrostatic, van der Waals and boundary coupling terms and differentiate for forces.
Symbols & assumptions
Additive QM/MM form; subtractive schemes use different bookkeeping. Link atoms, polarization and double counting require care.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Practical applications & products
Application area
Multiscale, reduced & data-driven models
Practical use
A local chemical event in a large molecular environment.
Problem & parameters. As a consistency check, choose a common harmonic coordinate whose total coupled-region energy is kq²/2 and whose effective mass is m.
q/A=cos(ωt)
Solution. The total force is −kq. Solve m q″+kq=0 with q(0)=A and q′(0)=0.
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Orange point: horizontal coordinate 3.1416, calculated vertical coordinate -1. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 3.1416, into the displayed formula to obtain -1 on the vertical axis. Values are rounded for display.
Scope. Prescribed harmonic reference only; no electronic calculation, interface force transfer, or adaptive region simulation is performed.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Connects particle-level and continuum descriptions.
Cross-scaleFramework
Mathematical model & short derivation
Representative formulation
Etotal≈Eatomistic+Econtinuum+Einterface
Derivation / construction sketch
Resolve atomistic detail where discrete effects matter.
Use an effective continuum energy away from that region.
Construct interface coupling that transfers forces and avoids double counting.
Symbols & assumptions
Schematic energy-based coupling; ghost forces and patch-test consistency depend on the specific method.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. As a consistency check, choose a common harmonic coordinate whose total coupled-region energy is kq²/2 and whose effective mass is m.
q/A=cos(ωt)
Solution. The total force is −kq. Solve m q″+kq=0 with q(0)=A and q′(0)=0.
Open this section to load its graph, or use the full-size example link below.
Orange point: horizontal coordinate 3.1416, calculated vertical coordinate -1. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 3.1416, into the displayed formula to obtain -1 on the vertical axis. Values are rounded for display.
Scope. Prescribed harmonic reference only; no electronic calculation, interface force transfer, or adaptive region simulation is performed.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Couples fluid loads with structural motion or deformation.
Cross-scaleFramework
Mathematical model & short derivation
Representative formulation
uf=∂tdsat Γσfnf+σsns=0
Derivation / construction sketch
Solve fluid and structural momentum equations in their respective domains.
Enforce matching interface velocity.
Enforce equal-and-opposite interface tractions and update geometry consistently.
Symbols & assumptions
No-slip fluid–structure coupling; partitioned or monolithic solvers need stable exchange and compatible interface discretization.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Approximate fluid loading as a constant added mass ma=m on an undamped spring-supported body.
q/A=cos(1+ma/mτ),ma/m=1
Solution. Combine the masses: (m+ma)q″+kq=0. The frequency becomes √[k/(m+ma)].
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Orange point: horizontal coordinate 6.2832, calculated vertical coordinate -0.26626. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 6.2832, into the displayed formula to obtain -0.26626 on the vertical axis. Values are rounded for display.
Scope. Linear added-mass reduction of FSI; no viscous drag, free-surface, or flow-field solution.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Couples temperature evolution and mechanical response.
Cross-scaleFramework
Mathematical model & short derivation
Representative formulation
σ=C:(ε−αΔTI)ρcpT˙=∇⋅(k∇T)+Q
Derivation / construction sketch
Represent thermal expansion as a stress-free strain.
Subtract it from total strain in the elastic constitutive law.
Solve the thermal energy balance and couple deformation-dependent heat or geometry effects when needed.
Symbols & assumptions
Linear isotropic thermoelastic example; α is thermal expansion coefficient. Full thermodynamics can include mechanical heating and reversible coupling.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. A one-dimensional elastic bar is prevented from expanding while its temperature rises uniformly.
σ/(EαΔT∗)=−ΔT/ΔT∗
Solution. Total strain is σ/E+αΔT. Set it to zero and solve for stress.
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Orange point: horizontal coordinate 1, calculated vertical coordinate -1. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 1, into the displayed formula to obtain -1 on the vertical axis. Values are rounded for display.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Builds a compact basis from representative field snapshots.
Cross-scaleFramework
Mathematical model & short derivation
Representative formulation
X=UΣVTx≈Ura
Derivation / construction sketch
Collect representative state snapshots into a matrix X.
Compute its singular value decomposition.
Keep the leading r left singular vectors, which minimize squared reconstruction error for an orthonormal rank-r basis.
Symbols & assumptions
POD alone provides a basis, not an evolution law; projecting dynamics or fitting reduced equations is an additional step.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Example 1: Exactly rank-one snapshot reconstruction
Exactly rank-one snapshot reconstruction
Problem & parameters. All snapshots are scalar multiples of sin(πx). Reconstruct the snapshot at dimensionless time one using one POD mode.
u(x,t)=e−tsin(πx),t=1
Solution. The snapshot matrix has rank one. Its only nonzero spatial mode is proportional to sin(πx), with coefficient exp(−t).
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Orange point: horizontal coordinate 0.5, calculated vertical coordinate 0.36788. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 0.36788 on the vertical axis. Values are rounded for display.
Scope. Exact rank-one constructed data set; real POD truncation can incur substantial error.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Projects a parameterized governing model onto a small approximation space.
Cross-scaleFramework
Mathematical model & short derivation
Representative formulation
VrTA(μ)Vrar=VrTb(μ)u≈Vrar
Derivation / construction sketch
Generate representative solutions over a parameter domain.
Build a small basis Vr from those solutions.
Project the governing equations onto that basis to reduce solve dimension.
Symbols & assumptions
Linear parameterized-system example; nonlinear problems need efficient evaluation or hyper-reduction and error control.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Practical applications & products
Application area
Multiscale, reduced & data-driven models
Practical use
Fast repeated evaluation of an engineering design.
Problem & parameters. Use uτ = uξξ on the unit interval, zero end values, and u(ξ,0) = sin(πξ). Plot τ = 0.1.
u(ξ,τ)=sin(πξ)e−π2τ,τ=0.1
Solution. The sine satisfies both zero end values. Its second derivative is −π² times itself; the amplitude solves a′ = −π²a.
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Orange point: horizontal coordinate 0.5, calculated vertical coordinate 0.37271. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 0.37271 on the vertical axis. Values are rounded for display.
Scope. Exact PDE benchmark. For reduced bases, PINNs, and neural operators, this is a reference target, not a claimed trained or computed prediction.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Predicts responses with a probabilistic function model fitted to samples.
Cross-scaleFramework
Mathematical model & short derivation
Representative formulation
μ∗=k∗T(K+σn2I)−1yσ∗2=k∗∗−k∗T(K+σn2I)−1k∗
Derivation / construction sketch
Assign a Gaussian-process prior with a chosen kernel.
Combine its joint Gaussian distribution at training and query points with a noise model.
Condition on observed data to obtain predictive mean and variance.
Symbols & assumptions
Zero-mean GP formulas; K is training covariance and k* cross-covariance. Predictive uncertainty depends on kernel and data assumptions.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Practical applications & products
Application area
Multiscale, reduced & data-driven models
Practical use
Emulating an expensive simulation over a bounded design space.
Example 1: One-observation Gaussian-process posterior mean
One-observation Gaussian-process posterior mean
Problem & parameters. Use a zero-mean, unit-variance squared-exponential GP, unit length scale, and one noiseless observation y(0)=1.
μ(x)=e−x2/2
Solution. The one-by-one training covariance is one. The conditional mean k(x,0)K^−1y equals exp(−x²/2).
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Orange point: horizontal coordinate 0, calculated vertical coordinate 1. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 1 on the vertical axis. Values are rounded for display.
Scope. Analytical posterior mean under the stated kernel; it is not a physical law, and posterior uncertainty is not shown.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Trains a neural approximation using data and governing-equation residuals.
Cross-scaleFramework
Mathematical model & short derivation
Representative formulation
L(θ)=λdata∑∣uθ−y∣2+λPDE∑∣N[uθ]−f∣2+λBCLBC
Derivation / construction sketch
Approximate the solution by a neural function uθ.
Evaluate governing-equation residuals and boundary errors, often using automatic differentiation.
Optimize a weighted loss combining physics and observations.
Symbols & assumptions
PINN method; a small sampled residual does not guarantee a uniformly accurate or conservative solution.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Practical applications & products
Application area
Multiscale, reduced & data-driven models
Practical use
An approximate solution of a specified differential equation.
Problem & parameters. Use uτ = uξξ on the unit interval, zero end values, and u(ξ,0) = sin(πξ). Plot τ = 0.1.
u(ξ,τ)=sin(πξ)e−π2τ,τ=0.1
Solution. The sine satisfies both zero end values. Its second derivative is −π² times itself; the amplitude solves a′ = −π²a.
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Orange point: horizontal coordinate 0.5, calculated vertical coordinate 0.37271. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 0.37271 on the vertical axis. Values are rounded for display.
Scope. Exact PDE benchmark. For reduced bases, PINNs, and neural operators, this is a reference target, not a claimed trained or computed prediction.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Learns a map between function-valued inputs and outputs.
Cross-scaleFramework
Mathematical model & short derivation
Representative formulation
u≈Gθ(a)vl+1(x)=σ[Wlvl(x)+∫κl(x,y)vl(y)dy]
Derivation / construction sketch
Represent the desired map from coefficient or forcing field a to solution field u.
Build layers that mix information locally and through an integral kernel.
Fit parameters from solution pairs, optionally adding physical constraints.
Symbols & assumptions
Representative neural-operator layer; Fourier operators parameterize the integral via spectral multipliers. Generalization requires validation.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Practical applications & products
Application area
Multiscale, reduced & data-driven models
Practical use
Surrogate prediction of parameterized solution fields.
Problem & parameters. Use uτ = uξξ on the unit interval, zero end values, and u(ξ,0) = sin(πξ). Plot τ = 0.1.
u(ξ,τ)=sin(πξ)e−π2τ,τ=0.1
Solution. The sine satisfies both zero end values. Its second derivative is −π² times itself; the amplitude solves a′ = −π²a.
Open this section to load its graph, or use the full-size example link below.
Orange point: horizontal coordinate 0.5, calculated vertical coordinate 0.37271. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 0.37271 on the vertical axis. Values are rounded for display.
Scope. Exact PDE benchmark. For reduced bases, PINNs, and neural operators, this is a reference target, not a claimed trained or computed prediction.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Assimilate measurements to update states or parameters.
Use the synchronized model for prediction and decisions, then continue updating as observations arrive.
Symbols & assumptions
Digital twin is an architecture, not a unique equation; identity, update cadence, uncertainty and validation are part of its definition.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Practical applications & products
Application area
Multiscale, reduced & data-driven models
Practical use
Updating a machine model using operating measurements.
Example 1: Digital-twin reference cooling trajectory
Digital-twin reference cooling trajectory
Problem & parameters. Use a lumped thermal model with a known constant cooling time as an ideal reference for a thermal digital twin.
(T−T∞)/(T0−T∞)=e−t/τ
Solution. Solve the first-order heat balance analytically; compare actual sensor data with this reference in a real implementation.
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Orange point: horizontal coordinate 2.5, calculated vertical coordinate 0.082085. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.082085 on the vertical axis. Values are rounded for display.
Scope. Reference physics only. No sensors, online updates, or actual equipment measurements are included.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Updates uncertain parameters using observations and a statistical likelihood.
Cross-scaleFramework
Mathematical model & short derivation
Representative formulation
p(θ∣y)∝p(y∣θ)p(θ)
Derivation / construction sketch
Choose a prior distribution for uncertain parameters.
Construct a likelihood from observations, noise and any model-discrepancy assumptions.
Apply Bayes’ rule to obtain a posterior for prediction and uncertainty propagation.
Symbols & assumptions
Identifiability and mismatch between model and reality can dominate; a narrow posterior does not prove model validity.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Practical applications & products
Application area
Multiscale, reduced & data-driven models
Practical use
Estimating heat-transfer parameters with uncertainty.
Problem & parameters. Use prior θ~Normal(0,1) and one measurement y=1 with independent Normal(0,1) measurement noise.
p(θ∣y=1)=π−1/2e−(θ−0.5)2
Solution. Add prior and data precisions to obtain variance 1/2; precision-weight the means to obtain posterior mean 1/2. Normalize the Gaussian.
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Orange point: horizontal coordinate 0.5, calculated vertical coordinate 0.56419. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 0.56419 on the vertical axis. Values are rounded for display.
Scope. Exact conjugate scalar calibration example; not a calibrated engineering system.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Represents uncertain responses with polynomial functions of random inputs.
Cross-scaleFramework
Mathematical model & short derivation
Representative formulation
Y(ξ)≈α∑cαΨα(ξ)cα=E[Ψα2]E[YΨα]
Derivation / construction sketch
Represent uncertainty with random inputs ξ.
Choose orthogonal polynomials for their probability law.
Project the response onto those polynomials or fit coefficients from samples.
Symbols & assumptions
Truncated polynomial chaos expansion; smoothness and dimension affect convergence. Correlated inputs require suitable transformations or bases.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Use a linear response to a uniform random input. Expand in the first two Legendre polynomials.
Y(ξ)=2+0.5ξ,ξ∼U[−1,1]
Solution. Since P0=1 and P1=ξ, coefficients are 2 and 0.5. The mean is 2 and variance is 0.25/3.
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Orange point: horizontal coordinate 0, calculated vertical coordinate 2. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 2 on the vertical axis. Values are rounded for display.
Scope. Exact degree-one expansion for the chosen response, not a surrogate fitted to arbitrary simulation data.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Match the dimensionless groups that control the phenomenon, rather than geometry alone.
Symbols & assumptions
A scale model cannot generally preserve all similarity conditions with the same fluid and gravity; prioritize the relevant physics.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Practical applications & products
Application area
Physical analogs & experimental models
Practical use
A reduced-size architectural or machinery prototype.
Problem & parameters. Scale all dimensions of a shape by the same positive length ratio.
Vm/Vp=(Lm/Lp)3
Solution. Volume is the product of three lengths; multiply the three identical scale factors.
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Orange point: horizontal coordinate 0.5, calculated vertical coordinate 0.125. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 0.125 on the vertical axis. Values are rounded for display.
Scope. Geometric similarity alone does not ensure force, material, or dynamic similarity.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Uses a controlled air stream around a physical specimen.
Cross-scalePhysical analog
Mathematical model & short derivation
Representative formulation
Rem=RepMam=Mapwhen both matter
Derivation / construction sketch
Identify viscous and compressibility effects through Reynolds and Mach numbers.
Choose model size, speed, fluid properties and pressure to match important nondimensional conditions.
Use nondimensional force coefficients to transfer measurements.
Symbols & assumptions
Wind-tunnel walls, support interference, transition and surface roughness also affect similarity.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Use fixed air density and reference dynamic pressure q*=ρU*²/2.
q/q∗=(U/U∗)2
Solution. Evaluate q=ρU²/2 and divide by q*.
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Orange point: horizontal coordinate 1, calculated vertical coordinate 1. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 1, into the displayed formula to obtain 1 on the vertical axis. Values are rounded for display.
Scope. Test-planning relation, not measured wind-tunnel data; Reynolds and Mach similarity require separate checks.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Uses physical water flow with selected similarity conditions.
Cross-scalePhysical analog
Mathematical model & short derivation
Representative formulation
Frm=FrpUprototypeUmodel=λL
Derivation / construction sketch
For gravity-dominated free-surface flow, match Froude number Fr=U/√(gL).
Use equal gravitational acceleration and a geometric scale λL.
Solve the equality for velocity scale and obtain time scale √λL.
Symbols & assumptions
Viscous and surface-tension similarity may conflict with Froude scaling, especially at small model sizes.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Use the same gravitational acceleration and match Froude number U/√(gL) between a model and prototype.
Um/Up=Lm/Lp
Solution. Equate the two Froude numbers and solve for the velocity ratio.
Open this section to load its graph, or use the full-size example link below.
Orange point: horizontal coordinate 0.505, calculated vertical coordinate 0.71063. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.505, into the displayed formula to obtain 0.71063 on the vertical axis. Values are rounded for display.
Scope. Gravity-dominated similarity appropriate to free-surface flumes; Reynolds, Weber, and other dimensionless groups may not also match.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Excites a physical structure with controlled base motion.
Cross-scalePhysical analog
Mathematical model & short derivation
Representative formulation
Mu¨+Cu˙+Ku=−Mrag(t)
Derivation / construction sketch
Write structure dynamics relative to a moving base.
Convert base acceleration ag into an equivalent inertial load.
Apply a scaled base-motion history in a shake-table experiment and compare response.
Symbols & assumptions
r is the influence vector. A physical test must match relevant mass, stiffness, damping and time scales.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Practical applications & products
Application area
Physical analogs & experimental models
Practical use
Investigating a scaled building's dynamic response.
Example 1: Undamped shake-table reference transfer
Undamped shake-table reference transfer
Problem & parameters. For an undamped single-degree-of-freedom oscillator with sinusoidal base motion, calculate the steady absolute displacement below resonance.
∣X/Y∣=1/∣1−r2∣,r=Ω/ωn
Solution. Insert harmonic motions into mX″+k(X−Y)=0. Solve (k−mΩ²)X=kY for the amplitude ratio.
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Orange point: horizontal coordinate 0.4, calculated vertical coordinate 1.1905. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.4, into the displayed formula to obtain 1.1905 on the vertical axis. Values are rounded for display.
Scope. Ideal steady reference, not shake-table measurements; the undamped resonance singularity is outside the plotted range.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Uses stress-induced optical birefringence to visualize stress patterns.
Cross-scalePhysical analog
Mathematical model & short derivation
Representative formulation
Nf=fσt(σ1−σ2)
Derivation / construction sketch
Use stress-induced birefringence to relate refractive-index difference to principal-stress difference.
Integrate optical retardation through specimen thickness t.
Express retardation as fringe order Nf using the calibrated stress-optic coefficient fσ.
Symbols & assumptions
Plane photoelasticity relation under suitable optical assumptions; calibration, residual stress and three-dimensional effects matter.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Practical applications & products
Application area
Physical analogs & experimental models
Practical use
Identifying stress concentrations in a transparent specimen.
Problem & parameters. Use a transparent specimen of thickness t, stress-optic coefficient C, and monochromatic wavelength λ.
N=Ct(σ1−σ2)/λ
Solution. The principal refractive-index difference is CΔσ. Optical path retardation is CtΔσ; divide by wavelength to obtain fringe order.
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Orange point: horizontal coordinate 2.5, calculated vertical coordinate 2.5. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 2.5 on the vertical axis. Values are rounded for display.
Scope. Uniform stress through thickness and linear stress-optic law; this is not a fringe photograph.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Maps another physical system onto an electrical network.
Cross-scalePhysical analog
Mathematical model & short derivation
Representative formulation
CthT˙+RthT−T∞=Q↔CV˙+V/R=I
Derivation / construction sketch
Write a lumped thermal storage-and-resistance balance.
Compare it term by term with Kirchhoff current balance for an RC circuit.
Map temperature to voltage and heat flow to current with chosen scale factors.
Symbols & assumptions
An electrical analog reproduces the mapped equations within component tolerances; it does not automatically reproduce all physical effects.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Use the scalar state equation y′+y=1 with y(0)=0, or transfer function 1/(s+1).
y(τ)=1−e−τ
Solution. The homogeneous response is Ce^−τ and the constant particular response is one. The initial state gives C=−1.
Open this section to load its graph, or use the full-size example link below.
Orange point: horizontal coordinate 2.5, calculated vertical coordinate 0.91792. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.91792 on the vertical axis. Values are rounded for display.
Scope. Exact linear plant reference. For bond graphs/electrical analogs use a single storage-and-resistance element; for HIL this is a reference trajectory, not measured hardware data.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Couples actual hardware to simulated parts of a system.
Cross-scalePhysical analog
Mathematical model & short derivation
Representative formulation
xs,k+1=Fd(xs,k,uh,k)yh,k=Hs(xs,k)
Derivation / construction sketch
Simulate the plant or missing subsystem in real time.
Exchange measured hardware outputs and simulated sensor signals through interfaces.
Advance the simulation within each hardware sampling deadline.
Symbols & assumptions
HIL coupling requires bounded latency, calibrated I/O and stability under discretization; equations depend on the simulated plant and real hardware.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Problem & parameters. Use the scalar state equation y′+y=1 with y(0)=0, or transfer function 1/(s+1).
y(τ)=1−e−τ
Solution. The homogeneous response is Ce^−τ and the constant particular response is one. The initial state gives C=−1.
Open this section to load its graph, or use the full-size example link below.
Orange point: horizontal coordinate 2.5, calculated vertical coordinate 0.91792. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.91792 on the vertical axis. Values are rounded for display.
Scope. Exact linear plant reference. For bond graphs/electrical analogs use a single storage-and-resistance element; for HIL this is a reference trajectory, not measured hardware data.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Uses dimensionless groups to relate tests across scales.
Cross-scalePhysical analog
Mathematical model & short derivation
Representative formulation
Πj=i∏xiaiji∑aijdim(xi)=0
Derivation / construction sketch
List the dimensional variables controlling a phenomenon.
Find exponent combinations whose length, mass, time and other dimensions cancel.
Relate the resulting dimensionless groups using theory or experiments.
Symbols & assumptions
Buckingham Π construction; n variables with a dimensional matrix of rank r give n−r independent groups, subject to completeness of the variable list.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Practical applications & products
Application area
Physical analogs & experimental models
Practical use
Matching Reynolds or Froude behavior where appropriate.
Problem & parameters. Use the same gravitational acceleration and match Froude number U/√(gL) between a model and prototype.
Um/Up=Lm/Lp
Solution. Equate the two Froude numbers and solve for the velocity ratio.
Open this section to load its graph, or use the full-size example link below.
Orange point: horizontal coordinate 0.505, calculated vertical coordinate 0.71063. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.505, into the displayed formula to obtain 0.71063 on the vertical axis. Values are rounded for display.
Scope. Gravity-dominated similarity appropriate to free-surface flumes; Reynolds, Weber, and other dimensionless groups may not also match.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Define the total correlation h=g-1 and direct correlation c.
Separate the correlation into a direct contribution and indirect chains through other particles.
Fourier transformation turns convolution into multiplication; solve for h_hat and use S=1+rho h_hat.
Symbols & assumptions
Homogeneous isotropic equilibrium fluid; rho is number density, g is radial distribution, k is wavevector. Fourier transform has no prefactor in the forward integral. A closure or supplied c is needed.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Practical applications & products
Application area
Liquid-state physics
Practical use
Interpreting scattering structure factors of liquid and colloidal samples.
Product / system examples
Small-angle scattering analysis of colloidal dispersions
Example 1: OZ structure factor with prescribed direct correlation
OZ structure factor with prescribed direct correlation
Problem & parameters. Assume rho times the Fourier-transformed direct correlation is −exp[−(kℓ)²]. Find S(k) from the OZ relation.
ρc(k)=−e−(kℓ)2,S(k)=1+e−(kℓ)21
Solution. Fourier transformation gives h_hat=c_hat/(1−rho c_hat). Substitute into S=1+rho h_hat to obtain the displayed expression. At k=0, S=1/2; at large k, S tends to 1.
Open this section to load its graph, or use the full-size example link below.
Orange point: horizontal coordinate 1.5, calculated vertical coordinate 0.90465. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 1.5, into the displayed formula to obtain 0.90465 on the vertical axis. Values are rounded for display.
Scope. A prescribed-correlation algebraic benchmark, not a self-consistent closure solution or measured scattering spectrum.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Closes the liquid integral equation using an approximate relation between pair correlations and interactions.
Atomic / molecularPhysical model
Mathematical model & short derivation
Representative formulation
c(r)=[e−βu(r)−1][1+γ(r)],γ=h−c,β=(kBT)−1
Derivation / construction sketch
Introduce the indirect correlation gamma=h-c.
Approximate g=exp(-beta u)(1+gamma), retaining a linear indirect-correlation factor.
Use c=g-1-gamma and solve the closure together with Ornstein-Zernike.
Symbols & assumptions
beta=1/(k_B T), u is pair energy. Approximate classical pair-potential theory; pressure routes can disagree. Hard cores require g=0 inside the core.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Practical applications & products
Application area
Liquid-state physics
Practical use
Fitting scattering from approximately neutral, hard-sphere colloidal dispersions.
Example 1: Percus-Yevick hard-sphere contact value
Percus-Yevick hard-sphere contact value
Problem & parameters. Use the analytical three-dimensional, monodisperse hard-sphere PY solution to evaluate its contact pair distribution as packing fraction varies.
g(σ+)=(1−ϕ)21+ϕ/2
Solution. The PY hard-sphere solution gives virial-route Z=(1+2φ+3φ²)/(1−φ)². The hard-sphere contact theorem Z=1+4φg(σ+) then gives g(σ+)=(1+φ/2)/(1−φ)² after subtraction and cancellation. At φ=0 use the limit g=1.
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Orange point: horizontal coordinate 0.225, calculated vertical coordinate 1.8522. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.225, into the displayed formula to obtain 1.8522 on the vertical axis. Values are rounded for display.
Scope. Contact-value evaluation of the PY approximation; the analytical OZ/PY solution is taken as the starting result. Thermodynamic routes are not identical.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Approximates liquid pair structure by neglecting bridge diagrams in the exact closure.
Atomic / molecularPhysical model
Mathematical model & short derivation
Representative formulation
g(r)=exp[−βu(r)+h(r)−c(r)],h(r)=g(r)−1
Derivation / construction sketch
Write the exact pair closure as g=exp(-beta u+gamma+B), with bridge contribution B.
Set B=0 to obtain the HNC approximation.
Combine h=g-1 with Ornstein-Zernike and iterate to self-consistency.
Symbols & assumptions
Classical equilibrium pair-potential fluid; beta=1/(k_B T). Bridge terms are omitted, which can impair dense, strongly correlated liquids. HNC is not an exact general liquid solution.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Practical applications & products
Application area
Liquid-state physics
Practical use
Estimating pair distributions in simple fluids with a specified pair potential.
Product / system examples
Custom liquid-structure calculations with Python/SciPy
Custom implementation route: supply the discretized OZ/HNC residual, transforms, and convergence checks. SciPy does not supply a built-in HNC liquid model.
Product categories above illustrate application areas; they do not assert an undocumented manufacturer’s design workflow.
Example, limitations & references
In practice
Estimating pair distributions in simple fluids with a specified pair potential.
Model-family limitations
Select an appropriate equilibrium, interaction, and transport regime. Closures and continuum limits have distinct validity ranges.
Example 1: Dilute HNC Gaussian-core pair distribution
Dilute HNC Gaussian-core pair distribution
Problem & parameters. Take the zero-density limit of an equilibrium soft Gaussian-core fluid with beta epsilon=1. Find its pair distribution.
βu(r)=e−(r/σ)2,g(r)=exp[−e−(r/σ)2]
Solution. As density tends to zero, OZ gives h=c and hence gamma=0. HNC reduces to the two-particle Boltzmann factor exp(−beta u); insert the specified Gaussian repulsion. At r=0, g=exp(−1).
Open this section to load its graph, or use the full-size example link below.
Orange point: horizontal coordinate 1.5, calculated vertical coordinate 0.89997. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 1.5, into the displayed formula to obtain 0.89997 on the vertical axis. Values are rounded for display.
Scope. Exact dilute two-particle limit for this specified potential; at finite liquid density, solve the coupled HNC/OZ equations instead.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Approximates the compressibility factor of a monodisperse hard-sphere fluid from its packing fraction.
Atomic / molecularPhysical model
Mathematical model & short derivation
Representative formulation
Z=ρkBTp=(1−ϕ)31+ϕ+ϕ2−ϕ3,ϕ=6πρσ3
Derivation / construction sketch
Define the occupied-volume fraction phi for spheres of diameter sigma.
Approximate virial coefficients by B_n=n^2+n-2 for n>=2 in the expansion in phi.
Sum the resulting geometric-series derivatives to obtain the rational expression for Z.
Symbols & assumptions
rho is number density, sigma sphere diameter, T temperature. Monodisperse, nonattracting hard-sphere fluid. This is a highly accurate approximation, not an exact EOS or a model of crystallization.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Practical applications & products
Application area
Liquid-state physics
Practical use
Estimating excluded-volume pressure contributions in dense-fluid reference models.
Product / system examples
Hard-sphere reference calculations for fluid-property research
Custom implementation route: evaluate the displayed algebraic expression. The reference is the original equation, not a claim of a built-in commercial EOS.
Product categories above illustrate application areas; they do not assert an undocumented manufacturer’s design workflow.
Example, limitations & references
In practice
Estimating excluded-volume pressure contributions in dense-fluid reference models.
Model-family limitations
Select an appropriate equilibrium, interaction, and transport regime. Closures and continuum limits have distinct validity ranges.
Problem & parameters. For a monodisperse hard-sphere fluid, compute pressure relative to ideal-gas pressure from packing fraction using Carnahan-Starling.
Z(ϕ)=(1−ϕ)31+ϕ+ϕ2−ϕ3
Solution. Insert φ into the numerator and denominator. For example, at φ=0.3 the numerator is 1.363 and denominator 0.343, giving Z=3.97376. The dilute limit is Z=1.
Open this section to load its graph, or use the full-size example link below.
Orange point: horizontal coordinate 0.225, calculated vertical coordinate 2.716. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.225, into the displayed formula to obtain 2.716 on the vertical axis. Values are rounded for display.
Scope. Constitutive evaluation of the approximate fluid EOS; no attractive forces, mixture effects, or solid phase are included.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Combines bracket reliability with interpolation-based acceleration.
Invert the hard-sphere EOS for packing fraction at a specified pressure using a bracket within the fluid regime. Direct pressure evaluation requires no root solver.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Substitute the drag coefficient to obtain the diffusion relation.
Symbols & assumptions
D is diffusivity, eta dynamic viscosity, R_H hydrodynamic radius. Dilute spherical probes in a Newtonian continuum solvent, low Reynolds number, no-slip boundary. Molecular-scale and concentrated systems may violate it.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Practical applications & products
Application area
Liquid-state physics
Practical use
Estimating hydrodynamic particle sizes from diffusion measurements in a dilute suspension.
Product / system examples
Diffusion-based particle sizing and Brownian-particle simulation
Example 1: Brownian sphere diffusion in a viscous solvent
Brownian sphere diffusion in a viscous solvent
Problem & parameters. Take T=298 K and solvent viscosity eta=0.001 Pa s. Estimate D for dilute no-slip spherical probes with radii between 10 and 200 nm.
D(R)=6π(10−3)(R×10−9)(1.380649×10−23)(298)m2s−1
Solution. Convert radius from nm to m and substitute into Stokes-Einstein. At R=100 nm, D=2.18273×10⁻¹² m²/s. Doubling radius halves diffusivity.
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Orange point: horizontal coordinate 105, calculated vertical coordinate 2.0788e-12. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 105, into the displayed formula to obtain 2.0788e-12 on the vertical axis. Values are rounded for display.
Scope. Chosen constant solvent viscosity, not a measured water-property curve. Continuum, no-slip, dilute-sphere assumptions apply.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Obtains equilibrium shear viscosity from the time integral of microscopic shear-stress fluctuations.
Atomic / molecularPhysical model
Mathematical model & short derivation
Representative formulation
η=kBTV∫0∞⟨δPxy(0)δPxy(t)⟩dt
Derivation / construction sketch
Linear response relates the shear response to equilibrium momentum-flux fluctuations.
Form the stationary autocorrelation of the off-diagonal intensive pressure tensor component P_xy.
Integrate the correlation and multiply by V/(k_B T); independent shear components can improve sampling.
Symbols & assumptions
Equilibrium isotropic liquid in volume V at T. P_xy is an intensive pressure (Pa), not a volume-integrated virial. Subtract any nonzero mean. Finite trajectories require convergence and tail-error checks.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Practical applications & products
Application area
Liquid-state physics
Practical use
Predicting liquid viscosity from an equilibrium molecular-dynamics trajectory.
Product / system examples
LAMMPS equilibrium viscosity calculations for simulated liquids
Example 1: Viscosity integral for an exponential stress correlation
Viscosity integral for an exponential stress correlation
Problem & parameters. Assume the equilibrium intensive shear-pressure autocorrelation C(t)=C0 exp(−t/τ), with C0>0. Calculate the running Green-Kubo viscosity integral.
C(t)=C0e−t/τ,η∞η(t)=1−e−t/τ,η∞=kBTVC0τ
Solution. Integrate C0 exp(−s/τ) from 0 to t to obtain C0τ[1−exp(−t/τ)]. Multiply by V/(kBT) and divide by its infinite-time limit. At t=3τ, 95.0213% of the assumed total is recovered.
Open this section to load its graph, or use the full-size example link below.
Orange point: horizontal coordinate 3, calculated vertical coordinate 0.95021. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 3, into the displayed formula to obtain 0.95021 on the vertical axis. Values are rounded for display.
Scope. Analytical exponential-correlation benchmark; real liquid stress correlations may oscillate or have long tails. This is not a molecular-dynamics measurement.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Advances positions and velocities with a symmetric force update.
Generate equilibrium MD trajectories and shear-pressure samples with a compatible ensemble and force field; this time integrator alone does not estimate viscosity.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Treats crystal vibrations as independent quantum oscillators at a single frequency.
Atomic / molecularPhysical model
Mathematical model & short derivation
Representative formulation
CV=3NkB(ex−1)2x2ex,x=TΘE,ΘE=kBℏωE
Derivation / construction sketch
Assign 3N identical quantum oscillators to N atoms.
Sum their mean energies using Bose occupation, including a temperature-independent zero-point term.
Differentiate energy with respect to T at fixed volume to obtain heat capacity.
Symbols & assumptions
Theta_E=hbar omega_E/k_B. Independent harmonic oscillators with one frequency; ignores dispersion, acoustic low-frequency modes, anharmonicity, and electronic heat capacity.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Practical applications & products
Application area
Solid-state physics
Practical use
Estimating an optical-mode contribution to crystal heat capacity.
Product / system examples
Cryogenic crystal calorimetry and optical-phonon heat-capacity fits
Custom single-frequency evaluation of the documented oscillator heat-capacity law. Phonopy normally uses a full phonon spectrum rather than an Einstein crystal.
Product categories above illustrate application areas; they do not assert an undocumented manufacturer’s design workflow.
Example, limitations & references
In practice
Estimating an optical-mode contribution to crystal heat capacity.
Model-family limitations
Idealized crystalline-solid models. Check dimensionality, temperature range, interactions, disorder, and parameter validity before material-specific use.
Problem & parameters. For 3N identical oscillators, calculate the normalized heat capacity versus temperature.
θ=T/ΘE,3NkBCV=(1−e−1/θ)2θ−2e−1/θ
Solution. Each oscillator has thermal energy hbar omega/[exp(hbar omega/kBT)−1]. Differentiate and divide the total by 3NkB. At T=ThetaE, the result is e/(e−1)²=0.920674.
Open this section to load its graph, or use the full-size example link below.
Orange point: horizontal coordinate 1.05, calculated vertical coordinate 0.92772. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 1.05, into the displayed formula to obtain 0.92772 on the vertical axis. Values are rounded for display.
Scope. Exact evaluation within the single-frequency harmonic Einstein model; the acoustic low-temperature cubic law is absent.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Regularizes a Gauss-Newton step to balance stability and progress.
Use fitting for an Einstein temperature, interpolation for tabulated responses, or refinement to check derived quantities; the displayed formula itself is explicit. For nonlinear least-squares fitting with damping; standard LM does not handle arbitrary constraints.
Joins piecewise cubic polynomials with continuity constraints.
Use fitting for an Einstein temperature, interpolation for tabulated responses, or refinement to check derived quantities; the displayed formula itself is explicit. For smooth interpolation of coefficients or responses; ordinary splines do not guarantee positivity or monotonicity.
Cancels a leading discretization-error term using two resolutions.
Use fitting for an Einstein temperature, interpolation for tabulated responses, or refinement to check derived quantities; the displayed formula itself is explicit. For systematically refined computations in an established asymptotic error regime; use consistent geometry and boundary data.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Assume three acoustic branches with linear dispersion and a density of modes proportional to omega².
Choose the Debye cutoff so the integral of the density of states contains 3N modes.
Integrate oscillator energies over this spectrum and differentiate at fixed volume.
Symbols & assumptions
Theta_D=hbar omega_D/k_B, N atom count. Isotropic harmonic continuum approximation with an effective sound speed; optical modes and anharmonic effects require extensions.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Practical applications & products
Application area
Solid-state physics
Practical use
Estimating low-temperature lattice heat capacity in crystalline solids.
Product / system examples
Low-temperature solid heat-capacity estimates for cryogenic components
Custom implementation of the Debye integral or its stated low-temperature limit; not a claim of a built-in materials package.
Product categories above illustrate application areas; they do not assert an undocumented manufacturer’s design workflow.
Example, limitations & references
In practice
Estimating low-temperature lattice heat capacity in crystalline solids.
Model-family limitations
Idealized crystalline-solid models. Check dimensionality, temperature range, interactions, disorder, and parameter validity before material-specific use.
Problem & parameters. In the regime T much smaller than ThetaD, estimate lattice heat capacity using the leading Debye asymptote.
NkBCV≃512π4(ΘDT)3
Solution. Extend the Debye integral upper limit to infinity. Its value is 4pi⁴/15, so multiplying by 9(T/ThetaD)³ gives 12pi⁴(T/ThetaD)³/5. Doubling temperature multiplies this leading term by eight.
Open this section to load its graph, or use the full-size example link below.
Orange point: horizontal coordinate 0.0275, calculated vertical coordinate 0.0048619. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.0275, into the displayed formula to obtain 0.0048619 on the vertical axis. Values are rounded for display.
Scope. Low-temperature analytical asymptote only; the plotted range stops at T/ThetaD=0.05. Use the finite-cutoff integral outside this regime.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Chooses nodes and weights to integrate high-degree polynomials efficiently.
Use quadrature for the full Debye integral and fitting for a Debye temperature; the cubic law is only a low-temperature asymptote. For smooth element, energy, or moment integrals; singular or discontinuous integrands need special rules.
Subdivides intervals according to local integration-error estimates.
Use quadrature for the full Debye integral and fitting for a Debye temperature; the cubic law is only a low-temperature asymptote. For deterministic low-dimensional integrals; identify singularities and verify error estimates.
Regularizes a Gauss-Newton step to balance stability and progress.
Use quadrature for the full Debye integral and fitting for a Debye temperature; the cubic law is only a low-temperature asymptote. For nonlinear least-squares fitting with damping; standard LM does not handle arbitrary constraints.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Fill free-electron momentum states with two spin states per wavevector up to the Fermi sphere.
State counting gives k_F=(3pi²n)^(1/3) and E_F=hbar²k_F²/(2m).
A Sommerfeld expansion at fixed electron number gives the leading heat capacity proportional to T.
Symbols & assumptions
N is electron count, n=N/V, T_F=E_F/k_B. Three-dimensional noninteracting spin-1/2 electrons with parabolic dispersion; the heat-capacity expression requires T much smaller than T_F.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Practical applications & products
Application area
Solid-state physics
Practical use
Estimating electronic heat capacity and Fermi energy in simple metals.
Product / system examples
Normal-metal electronic calorimetry and simple-metal reference calculations
Custom implementation of the ideal free-electron reference equations; real band structure and many-body renormalization require extensions.
Product categories above illustrate application areas; they do not assert an undocumented manufacturer’s design workflow.
Example, limitations & references
In practice
Estimating electronic heat capacity and Fermi energy in simple metals.
Model-family limitations
Idealized crystalline-solid models. Check dimensionality, temperature range, interactions, disorder, and parameter validity before material-specific use.
Problem & parameters. Find the leading electronic heat capacity of a three-dimensional free-electron gas at fixed electron number and low temperature.
NkBCe≃2π2TFT
Solution. The fixed-number Sommerfeld expansion gives U/N=(3/5)EF+(pi²/4)(kBT)²/EF to this order. Differentiate with respect to temperature, and use EF=kB TF. At T/TF=0.01, Ce/(NkB)=0.0493480.
Open this section to load its graph, or use the full-size example link below.
Orange point: horizontal coordinate 0.0255, calculated vertical coordinate 0.12584. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.0255, into the displayed formula to obtain 0.12584 on the vertical axis. Values are rounded for display.
Scope. Leading low-temperature contribution of ideal electrons only; excludes lattice heat capacity, band corrections, interactions, and superconductivity.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Chooses nodes and weights to integrate high-degree polynomials efficiently.
Integrate the free-electron density of states with Fermi occupations at finite temperature, or fit a low-temperature heat-capacity coefficient; preserve electron number. For smooth element, energy, or moment integrals; singular or discontinuous integrands need special rules.
Subdivides intervals according to local integration-error estimates.
Integrate the free-electron density of states with Fermi occupations at finite temperature, or fit a low-temperature heat-capacity coefficient; preserve electron number. For deterministic low-dimensional integrals; identify singularities and verify error estimates.
Regularizes a Gauss-Newton step to balance stability and progress.
Integrate the free-electron density of states with Fermi occupations at finite temperature, or fit a low-temperature heat-capacity coefficient; preserve electron number. For nonlinear least-squares fitting with damping; standard LM does not handle arbitrary constraints.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Expand electronic states in localized orbitals centered on lattice sites.
Retain on-site energies and selected hopping matrix elements.
For a uniform one-dimensional nearest-neighbor chain, insert a Bloch wave to obtain the cosine band.
Symbols & assumptions
Displayed chain uses an orthonormal single orbital per site and real positive hopping t. Multiorbital materials require calibrated matrix elements and possibly overlap and spin-orbit coupling. Electron correlations are excluded unless added explicitly.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Practical applications & products
Application area
Solid-state physics
Practical use
Modeling semiconductor nanowire and crystalline-device electronic bands.
Product / system examples
QuantumATK Slater-Koster calculations for silicon nanowires
The vendor documents silicon nanowire calculations with parameterized orbital models; the displayed one-orbital chain is an educational specialization.
Product categories above illustrate application areas; they do not assert an undocumented manufacturer’s design workflow.
Example, limitations & references
In practice
Modeling semiconductor nanowire and crystalline-device electronic bands.
Model-family limitations
Idealized crystalline-solid models. Check dimensionality, temperature range, interactions, disorder, and parameter validity before material-specific use.
Problem & parameters. Use a one-dimensional chain with one orbital per site and positive nearest-neighbor hopping t. Find its band over half the Brillouin zone.
tE(k)−ϵ0=−2cos(ka)
Solution. Insert amplitudes c_n=exp(ikna) into E c_n=epsilon0 c_n−t(c_(n+1)+c_(n−1)). Divide by c_n and combine the two exponentials as 2cos(ka). The band runs from epsilon0−2t to epsilon0+2t.
Open this section to load its graph, or use the full-size example link below.
Orange point: horizontal coordinate 1.5708, calculated vertical coordinate -1.2246e-16. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 1.5708, into the displayed formula to obtain -1.2246e-16 on the vertical axis. Values are rounded for display.
Scope. One-orbital, orthonormal, noninteracting chain. The other half-zone follows by inversion symmetry; real semiconductor bands generally need multiple orbitals.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Separates matrix directions by their amplification strengths.
Use SVD on H(k)-E I to check null states, condition analysis to assess sensitivity, and quadrature for Brillouin-zone averages. General bands require a Hermitian eigensolver, not SVD singular values interpreted as signed energies. For reduced bases, rank diagnosis, or regularized inverse fitting; select truncation using the data and error budget.
Measures how perturbations in inputs can affect a computed solution.
Use SVD on H(k)-E I to check null states, condition analysis to assess sensitivity, and quadrature for Brillouin-zone averages. General bands require a Hermitian eigensolver, not SVD singular values interpreted as signed energies. For assembled linear systems or fitting matrices; separate problem conditioning from algorithm stability.
Chooses nodes and weights to integrate high-degree polynomials efficiently.
Use SVD on H(k)-E I to check null states, condition analysis to assess sensitivity, and quadrature for Brillouin-zone averages. General bands require a Hermitian eigensolver, not SVD singular values interpreted as signed energies. For smooth element, energy, or moment integrals; singular or discontinuous integrands need special rules.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Expand the periodic lattice potential in reciprocal-lattice Fourier components.
Near a Bragg degeneracy, keep the two coupled plane waves k and k-G.
Diagonalize their 2 by 2 Hamiltonian; at exact degeneracy the energy gap is 2 abs(V_G).
Symbols & assumptions
epsilon_k=hbar²k²/(2m); a mean potential can be absorbed into the energy origin. Weak periodic potential, two-state approximation near a Bragg plane; remote states and strong correlations are omitted.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Practical applications & products
Application area
Solid-state physics
Practical use
Explaining Bragg-plane band gaps in weak-potential crystalline conductors.
Product / system examples
Custom weak-periodic-potential electronic band calculations
Custom implementation route: assemble a Hermitian plane-wave Hamiltonian and diagonalize it. NumPy supplies linear algebra, not a built-in solid-state model.
Product categories above illustrate application areas; they do not assert an undocumented manufacturer’s design workflow.
Example, limitations & references
In practice
Explaining Bragg-plane band gaps in weak-potential crystalline conductors.
Model-family limitations
Idealized crystalline-solid models. Check dimensionality, temperature range, interactions, disorder, and parameter validity before material-specific use.
Solution. The two free plane-wave energies are ER(1+q)² and ER(q−1)². Diagonalizing their 2 by 2 matrix gives the displayed lower eigenvalue. At q=0 the energies are 0.9 ER and 1.1 ER, separated by 0.2 ER.
Open this section to load its graph, or use the full-size example link below.
Orange point: horizontal coordinate 0.15, calculated vertical coordinate 0.70627. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0.15, into the displayed formula to obtain 0.70627 on the vertical axis. Values are rounded for display.
Scope. Exact two-state diagonalization, approximate nearly-free-electron physics. Only the lower branch on one side of the Bragg plane is plotted; remote plane waves are omitted.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Separates matrix directions by their amplification strengths.
Use SVD for null-state diagnostics, sensitivity checks for small gaps, and quadrature for band averages. Diagonalize the Hermitian plane-wave Hamiltonian for actual energies. For reduced bases, rank diagnosis, or regularized inverse fitting; select truncation using the data and error budget.
Measures how perturbations in inputs can affect a computed solution.
Use SVD for null-state diagnostics, sensitivity checks for small gaps, and quadrature for band averages. Diagonalize the Hermitian plane-wave Hamiltonian for actual energies. For assembled linear systems or fitting matrices; separate problem conditioning from algorithm stability.
Chooses nodes and weights to integrate high-degree polynomials efficiently.
Use SVD for null-state diagnostics, sensitivity checks for small gaps, and quadrature for band averages. Diagonalize the Hermitian plane-wave Hamiltonian for actual energies. For smooth element, energy, or moment integrals; singular or discontinuous integrands need special rules.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Expand crystal potential energy to second order in atomic displacements about equilibrium.
Fourier-transform the force-constant equations and mass-weight them to form the dynamical matrix.
Diagonalize the matrix at each wavevector to obtain squared phonon frequencies and polarizations.
Symbols & assumptions
D is the mass-weighted dynamical matrix; e is polarization, nu branch index. The displayed one-dimensional chain has nearest-neighbor spring constant K, mass m, and spacing a. Harmonic approximation excludes phonon scattering and thermal expansion.
For empirical models, the sketch explains construction or fitting rather than a first-principles derivation. See the entry’s references for the complete formulation.
Practical applications & products
Application area
Solid-state physics
Practical use
Predicting phonon dispersion, harmonic stability, and vibrational thermodynamics.
Product / system examples
Phonopy phonon spectra and harmonic thermodynamic calculations
Documented dynamical-matrix and thermodynamic implementation. Material force constants are required from an appropriate external calculation or fitted model.
Product categories above illustrate application areas; they do not assert an undocumented manufacturer’s design workflow.
Example, limitations & references
In practice
Predicting phonon dispersion, harmonic stability, and vibrational thermodynamics.
Model-family limitations
Idealized crystalline-solid models. Check dimensionality, temperature range, interactions, disorder, and parameter validity before material-specific use.
Problem & parameters. Take identical masses m separated by a, joined by nearest-neighbor springs K. Find the normal-mode dispersion over half the Brillouin zone.
2K/mω(q)=sin2qa(0≤qa≤π)
Solution. Insert u_n=A exp[i(qna−omega t)] into m u_n″=K(u_(n+1)+u_(n−1)−2u_n). This gives omega²=(4K/m)sin²(qa/2). Select the nonnegative frequency. At small q the sound speed is a sqrt(K/m).
Open this section to load its graph, or use the full-size example link below.
Orange point: horizontal coordinate 1.5708, calculated vertical coordinate 0.70711. Axis labels specify the quantities and units or normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 1.5708, into the displayed formula to obtain 0.70711 on the vertical axis. Values are rounded for display.
Scope. One-dimensional harmonic monatomic chain; no optical branch, anharmonic scattering, or measured material parameters.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Conditional starting points, not automatic solver selections. Check the actual equations, constraints, stiffness, and matrix structure. Some linked atlas entries are methods or frameworks rather than physical laws. See the linked entries’ assumptions and references.
Separates matrix directions by their amplification strengths.
Fit force constants with SVD, cross-check real-time harmonic motion with Verlet, and test displacement/time-step refinement. Obtain phonon frequencies with a Hermitian dynamical-matrix eigensolver. For reduced bases, rank diagnosis, or regularized inverse fitting; select truncation using the data and error budget.
Advances positions and velocities with a symmetric force update.
Fit force constants with SVD, cross-check real-time harmonic motion with Verlet, and test displacement/time-step refinement. Obtain phonon frequencies with a Hermitian dynamical-matrix eigensolver. For compatible position-dependent conservative forces; constraints, thermostats, and stochastic forces require specialized extensions.
Cancels a leading discretization-error term using two resolutions.
Fit force constants with SVD, cross-check real-time harmonic motion with Verlet, and test displacement/time-step refinement. Obtain phonon frequencies with a Hermitian dynamical-matrix eigensolver. For systematically refined computations in an established asymptotic error regime; use consistent geometry and boundary data.
Editorial connections and subject grouping, not a complete dependency graph. Assumptions matter; consult the linked entries’ formulations and references.
Choose the simplest model that answers your question.
01
Define the outcome
Choose the quantities, length and time scales, operating conditions, and accuracy you need. A molecular trajectory and a system-level estimate answer different questions.
02
Check the assumptions
Decide which physics matter. Check continuum assumptions, equilibrium, linearity, material laws, boundary conditions, and whether multiple scales must be coupled.
03
Verify, then validate
Check units, conservation, numerical convergence and limiting cases. Compare predictions with independent measurements and report uncertainty and model limitations.
ABOUT THIS ATLAS
A starting point for discovery.
Modeling brings together physical laws, constitutive models, empirical approximations, experimental analogs, numerical methods, and modeling frameworks. Entry types keep these roles visible: a finite element method, for example, is a way to solve a model rather than a physical law.
This release includes 269 entries across 25 subject areas. Named families may contain many variants; software names appear in references, not as claims that all implementations are equivalent. New and specialist models continue to emerge.
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