m physical modeling / IICSM

CONNECTED PHYSICS / STATES OF MATTER

Gaseous-state physics

Use diluteness to separate free flight from binary collisions. From a velocity distribution, derive pressure and fluid equations, then introduce ionization when the species composition changes.

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Matter pathway: atom → solid → liquid → gas → plasma. Quantum mechanics and quantum field theory provide foundations across the pathway; they are not additional phases. This is a connected modeling route, not a universal heating curve. Actual phases depend on pressure, composition, and kinetics.

Jump to gas / liquid / solid comparison plots ↓

Compare gas, liquid, and solid structure

The same colors identify each reference model in all three plots. σ is a reference length: the liquid hard-sphere diameter and the solid nearest-neighbor lattice spacing. All functions are dimensionless; S(k) uses a logarithmic vertical axis so both low-k suppression and sharp peaks remain visible.

Gas has g=1. The liquid has an excluded core and a damped sequence of neighbor shells. The crystal retains a series of sharper lattice-shell peaks. Labeled axes compare ideal gas, hard-sphere liquid, and Gaussian FCC solid.
Gas has g=1. The liquid has an excluded core and a damped sequence of neighbor shells. The crystal retains a series of sharper lattice-shell peaks. Full-size plot ↗ · Download SVG
Subtract one from each g(r) curve. Positive regions mean excess pair density; negative regions mean depletion. The ideal-gas curve is zero. Labeled axes compare ideal gas, hard-sphere liquid, and Gaussian FCC solid.
Subtract one from each g(r) curve. Positive regions mean excess pair density; negative regions mean depletion. The ideal-gas curve is zero. Full-size plot ↗ · Download SVG
Gas has S=1. The liquid has a broad diffraction maximum. The crystal has narrow, resolution-broadened Bragg peaks above diffuse scattering; the forward k=0 beam is omitted. Labeled axes compare ideal gas, hard-sphere liquid, and Gaussian FCC solid.
Gas has S=1. The liquid has a broad diffraction maximum. The crystal has narrow, resolution-broadened Bragg peaks above diffuse scattering; the forward k=0 beam is omitted. Full-size plot ↗ · Download SVG
Models, parameters, and how the curves were calculated
  • Gas: ideal uncorrelated classical particles: g(r)=1, h(r)=0, S(k)=1 at nonzero wavevector.
  • Liquid: three-dimensional monodisperse hard spheres at packing fraction φ=0.40, nσ³=6φ/π≈0.76394. The Percus–Yevick direct-correlation polynomial is Fourier-transformed, then S=1/(1−nĉ). The radial inverse of S−1 gives h outside the core. g=0 for r<σ and the exact PY right-contact value g(σ+)=3.33333 are imposed at the discontinuity. Inversion uses kmaxσ=400 and Δkσ=0.05; small truncation oscillations remain near contact.
  • Solid: an FCC point lattice with independent isotropic Gaussian displacements of standard deviation 0.045σ per Cartesian coordinate; nσ³=√2. Radial g is the powder average of Gaussian relative-displacement distributions summed over lattice neighbors. h is exactly g−1. This harmonic reference allows arbitrarily close pairs with extremely small probability; it is not a hard-sphere solid.
  • Crystal S(k): diffuse scattering 1−exp(−k²s²), plus FCC reciprocal-lattice Bragg shells weighted by exp(−G²s²). Each radial delta peak is replaced by an area-normalized Gaussian of width 0.12 in kσ for display. The intensity includes elastic Bragg scattering; the uniform forward peak at k=0 is excluded. It is not the connected fluctuation spectrum about a fixed periodic mean density.

The curves use different densities and statistical models. They illustrate order, not a single material at one temperature or a simulated phase-change trajectory. The solid is powder-averaged; a single crystal has direction-dependent diffraction. Do not apply the homogeneous-liquid compressibility formula to the displayed crystal Bragg spectrum, or infer pressure from a peak height. The display-broadened crystal S is not an exact inverse transform of the unsmoothed radial g shown here.

SasView: PY hard-sphere reference ↗ · LAMMPS: crystalline diffraction definitions ↗

Reproducible plot data: g(r) and h(r) CSV · S(k) CSV · All data and parameters (JSON)

1. Factorize the dilute-gas partition function

λT=h2πmkBT,ZN=1N!(VλT3)N\lambda_T=\frac{h}{\sqrt{2\pi m k_{\mathrm B}T}},\quad Z_N=\frac1{N!}\left(\frac{V}{\lambda_T^3}\right)^NF≃NkBT[ln⁡(nλT3)−1]F\simeq Nk_{\mathrm B}T[\ln(n\lambda_T^3)-1]p=nkBT,U=32NkBTp=nk_{\mathrm B}T,\quad U=\frac32Nk_{\mathrm B}T
  1. Neglect interparticle potential energy for an ideal monatomic gas. The position integral gives V^N and the Gaussian momentum integral supplies the thermal wavelength.
  2. Apply Stirling’s approximation to ln(N!) in the thermodynamic limit to obtain F.
  3. Differentiate F with respect to V for pressure; average three translational quadratic degrees of freedom for U.

Connection / worked consequence. Worked example: N=10²³ particles, V=0.01 m³, T=300 K give p=41,419.47 Pa and U=621.292 J. These are ideal monatomic estimates, not universal gas properties.

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2. From correlated particles to binary collisions

∂tf+v⋅∇rf+Fextm⋅∇vf=CB[f]\partial_t f+\mathbf v\cdot\nabla_{\mathbf r}f+\frac{\mathbf F_{\mathrm{ext}}}{m}\cdot\nabla_{\mathbf v}f=C_B[f]CB[f]=∫d3v1 dΩ greldσdΩ[f′f1′−ff1]C_B[f]=\int d^3v_1\,d\Omega\,g_{\mathrm{rel}}\frac{d\sigma}{d\Omega}\left[f^{\prime}f_1^{\prime}-ff_1\right]
  1. Start from the reduced-distribution hierarchy on the overview page. Resolve encounters as brief, isolated binary collisions between long free flights.
  2. Approximate incoming two-particle velocity statistics by a product of one-particle distributions: molecular chaos.
  3. Count gain and loss into a velocity element using the differential cross-section, relative speed, and collision kinematics. For reversible elastic collisions the primed variables can label the corresponding inverse process.

Connection / worked consequence. Conservation of particle number, momentum, and energy in each elastic collision produces the corresponding collision-integral invariants.

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3. Velocity moments produce fluid balance laws

n=∫f d3v,nu=∫vf d3v,ρ=mnn=\int f\,d^3v,\quad n\mathbf u=\int\mathbf v f\,d^3v,\quad\rho=mn∂tρ+∇⋅(ρu)=0\partial_t\rho+\nabla\cdot(\rho\mathbf u)=0ρDuDt=−∇⋅P+nFext\rho\frac{D\mathbf u}{Dt}=-\nabla\cdot\mathsf P+n\mathbf F_{\mathrm{ext}}P=m∫(v−u)(v−u)f d3v\mathsf P=m\int(\mathbf v-\mathbf u)(\mathbf v-\mathbf u)f\,d^3v
  1. Integrate the kinetic equation over velocity; the number-conserving collision integral vanishes, giving continuity.
  2. Multiply by m v and integrate. Separate bulk motion from peculiar velocity to obtain momentum balance and the pressure tensor.
  3. A local Maxwellian gives P=pI and Euler-level closure. For small Knudsen number and near local equilibrium, a first-order Chapman-Enskog treatment generates viscosity and thermal conductivity for Navier-Stokes-Fourier.

Connection / worked consequence. Large Kn, strong nonequilibrium, or sharp layers can require Boltzmann or DSMC rather than conventional continuum closure.

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4. Gas → plasma: add reaction equilibrium or kinetics

A⇌A++e−,μA=μA++μeA\rightleftharpoons A^++e^-,\quad\mu_A=\mu_{A^+}+\mu_enenin0=2Ui(T)U0(T)(2πmekBTh2)3/2e−χ/(kBT)≡S(T)\frac{n_en_i}{n_0}=\frac{2U_i(T)}{U_0(T)}\left(\frac{2\pi m_ek_{\mathrm B}T}{h^2}\right)^{3/2}e^{-\chi/(k_{\mathrm B}T)}\equiv S(T)
  1. Allow ionization to change the populations of neutral atoms, ions, and electrons while conserving nuclei and charge.
  2. For local thermodynamic equilibrium, equate the reaction chemical potentials and use ideal nondegenerate translational partition functions.
  3. Separate the ionization energy χ and internal partition functions U0, Ui; the free-electron spin factor gives the displayed Saha relation. Away from equilibrium, use ionization/recombination rate equations instead.

Connection / worked consequence. Ionization can be partial and gradual. A plasma also requires collective electromagnetic behavior on the scales of interest; it is not defined by a single universal temperature.

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Related models & worked graphical examples

These entries come from the existing catalog. Open a formulation here, or follow its link for the complete model and three worked examples.

Ideal gas equation of state

Relates pressure, volume and temperature for a dilute noninteracting gas.

pV=nRTpV=nRT

Assumptions. n is amount in moles, V volume and T absolute temperature; intermolecular interactions and finite molecular volume are neglected.

Application. Estimating the amount of air in a low-pressure vessel.

Full model, derivation, references, numerical techniques & 3 worked graphs ↗
Van der Waals equation of state

Adds molecular attraction and excluded volume to an ideal gas model.

(p+a/v2)(v−b)=RT(p+a/v^2)(v-b)=RT

Assumptions. v is molar volume; a and b are substance parameters. This is a qualitative equation of state near critical and coexistence regions.

Application. Qualitative liquid-vapor coexistence.

Full model, derivation, references, numerical techniques & 3 worked graphs ↗
Virial equation of state

Represents nonideal behavior as a density or pressure expansion.

Z=pvRT=1+B(T)v+C(T)v2+⋯Z=\frac{pv}{RT}=1+\frac{B(T)}v+\frac{C(T)}{v^2}+\cdots

Assumptions. B and C are temperature-dependent molar virial coefficients; this low-density expansion may converge poorly near condensation.

Application. Gas properties away from the dilute limit.

Full model, derivation, references, numerical techniques & 3 worked graphs ↗
Navier–Stokes model

Conserves mass and momentum for a viscous continuum fluid.

ρ(∂tu+u⋅∇u)=−∇p+μ∇2u+ρg\rho(\partial_tu+u\cdot\nabla u)=-\nabla p+\mu\nabla^2u+\rho g∇⋅u=0\nabla\cdot u=0

Assumptions. u is velocity, p pressure, ρ density and μ dynamic viscosity. Compressible flow also needs energy and an equation of state.

Application. Water flow around a valve.

Full model, derivation, references, numerical techniques & 3 worked graphs ↗
Euler flow model

Neglects viscous stresses in compressible or incompressible flow.

ρDuDt=−∇p+ρg\rho\frac{Du}{Dt}=-\nabla p+\rho g∂tρ+∇⋅(ρu)=0\partial_t\rho+\nabla\cdot(\rho u)=0

Assumptions. D/Dt = ∂t+u·∇ is the material derivative. Inviscid approximations do not reproduce no-slip wall layers.

Application. First estimates of inviscid aerodynamic behavior.

Full model, derivation, references, numerical techniques & 3 worked graphs ↗
Boltzmann kinetic equation

Evolves a particle distribution under transport and collisions.

∂tf+v⋅∇xf+Fm⋅∇vf=C[f]\partial_tf+v\cdot\nabla_xf+\frac Fm\cdot\nabla_vf=C[f]

Assumptions. Boltzmann kinetic equation; collision assumptions, molecular interaction laws and closure determine C[f].

Application. Gas kinetics outside simple continuum conditions.

Full model, derivation, references, numerical techniques & 3 worked graphs ↗
Direct simulation Monte Carlo (DSMC)

Samples particle motion and collisions in a rarefied gas.

Ppair∝σT(g)gΔtVcellP_{\mathrm{pair}}\propto\frac{\sigma_T(g)g\Delta t}{V_{\mathrm{cell}}}

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.

Application. Gas flow where continuum assumptions fail.

Full model, derivation, references, numerical techniques & 3 worked graphs ↗

Notation used throughout

n: number density (m⁻³); N: particle count; ρ: mass density (kg m⁻³); ρc: charge density; p: pressure; T: temperature (K); kB: Boltzmann constant; h, ℏ: Planck constants; β=1/(kBT); μ: chemical potential; f: phase-space distribution; g(r): pair distribution. In solid displacements u is a displacement; in the liquid closure u(r) is pair energy; in fluid equations u is bulk velocity. Subscripts identify phase or species. Every approximation must use consistent SI units or explicitly stated reduced units.