m physical modeling / IICSM

CONNECTED PHYSICS / STATES OF MATTER

Plasma physics

Retain mobile charge and self-consistent fields. Choose collisional kinetics, collisionless kinetics, or a fluid approximation by comparing physical length and time scales.

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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.

1. Solve ionization together with charge and nuclei conservation

ni=ne=xnnuc,n0=(1−x)nnucn_i=n_e=xn_{\mathrm{nuc}},\quad n_0=(1-x)n_{\mathrm{nuc}}x21−x=S(T)nnuc≡A\frac{x^2}{1-x}=\frac{S(T)}{n_{\mathrm{nuc}}}\equiv Ax=2AA+A2+4A(A>0)x=\frac{2A}{A+\sqrt{A^2+4A}}\quad(A>0)
  1. For a single atomic species with only the first ionization stage, impose bulk charge neutrality and conservation of nuclei.
  2. Substitute these populations into Saha’s relation from the gas page to obtain x²=A(1−x).
  3. Choose the root between zero and one. The displayed rationalized form avoids cancellation at large A; for A=0 the solution is x=0.

Connection / worked consequence. Worked example: A=1 gives x=(√5−1)/2≈0.618034. This is a dimensionless equilibrium example, not a universal temperature for 62% ionization.

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2. Derive collective screening and its validity criteria

ns(ϕ)=ns0e−qsϕ/(kBTs)≃ns0(1−qsϕkBTs)n_s(\phi)=n_{s0}e^{-q_s\phi/(k_{\mathrm B}T_s)}\simeq n_{s0}\left(1-\frac{q_s\phi}{k_{\mathrm B}T_s}\right)∇2ϕ=λD−2ϕ,λD−2=∑sns0qs2ϵ0kBTs\nabla^2\phi=\lambda_D^{-2}\phi,\quad\lambda_D^{-2}=\sum_s\frac{n_{s0}q_s^2}{\epsilon_0k_{\mathrm B}T_s}ϕ(r)=Q4πϵ0re−r/λD\phi(r)=\frac{Q}{4\pi\epsilon_0r}e^{-r/\lambda_D}
  1. Use an isothermal Boltzmann response for each mobile species in a small electrostatic potential.
  2. Expand to first order and cancel the unperturbed charge density by neutrality. Insert the induced charge into Poisson’s equation, away from the test charge.
  3. Solve the radial screened-Poisson equation with decay at infinity; the source singularity fixes the Coulomb prefactor.

Connection / worked consequence. For a classical collective plasma, check λD/L much smaller than one and many particles in a Debye sphere. Also compare plasma oscillation, collision, and observation times. Ionization fraction alone is insufficient.

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3. Keep the mean field, then decide whether collisions matter

∂tfs+v⋅∇fs+qsms(E+v×B)⋅∇vfs=Cs[f]\partial_t f_s+\mathbf v\cdot\nabla f_s+\frac{q_s}{m_s}(\mathbf E+\mathbf v\times\mathbf B)\cdot\nabla_v f_s=C_s[f]ρc=∑sqs∫fs d3v,J=∑sqs∫vfs d3v\rho_c=\sum_s q_s\int f_s\,d^3v,\quad\mathbf J=\sum_s q_s\int\mathbf v f_s\,d^3v∇⋅E=ρc/ϵ0,∇⋅B=0\nabla\cdot\mathbf E=\rho_c/\epsilon_0,\quad\nabla\cdot\mathbf B=0∂tB=−∇×E,∇×B=μ0J+μ0ϵ0∂tE\partial_t\mathbf B=-\nabla\times\mathbf E,\quad\nabla\times\mathbf B=\mu_0\mathbf J+\mu_0\epsilon_0\partial_t\mathbf E
  1. Evolve each charged species under its Lorentz force, using fields sourced by the distributions themselves.
  2. When collisional changes are negligible over the time of interest, set Cs=0 to obtain Vlasov-Maxwell. Collisionless does not mean particles feel no forces.
  3. In an electrostatic approximation, set E=−grad φ and solve Poisson’s equation; omitting magnetic and radiative dynamics gives a Vlasov-Poisson specialization.

Connection / worked consequence. Weak long-range interactions can accumulate into a large collective field even when individual encounters are small. This is why ideal neutral-gas kinetics cannot simply be reused unchanged.

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4. Take moments and close at magnetohydrodynamic scales

ρDuDt=−∇p+J×B\rho\frac{D\mathbf u}{Dt}=-\nabla p+\mathbf J\times\mathbf BE+u×B=ηelJ\mathbf E+\mathbf u\times\mathbf B=\eta_{\mathrm{el}}\mathbf J∂tB=∇×(u×B)+ηelμ0∇2B,∇⋅B=0\partial_t\mathbf B=\nabla\times(\mathbf u\times\mathbf B)+\frac{\eta_{\mathrm{el}}}{\mu_0}\nabla^2\mathbf B,\quad\nabla\cdot\mathbf B=0
  1. Take species velocity moments, sum momentum equations, and adopt a single-fluid bulk velocity with an appropriate total-pressure closure.
  2. Neglect electron inertia, Hall, and other generalized-Ohm terms under the selected ordering; use a scalar electrical resistivity ηel.
  3. Combine Ohm’s law with Faraday’s law and Ampere’s law without displacement current. For uniform resistivity and div B=0, obtain the displayed induction equation.

Connection / worked consequence. Ideal MHD additionally sets ηel=0. A plasma may instead be collisionless and anisotropic, requiring kinetic or extended-fluid closures; plasma does not automatically imply MHD.

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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.

Debye–Hückel model

Approximates ionic activity using screened electrostatic interactions.

log⁡10γi=−Azi2I\log_{10}\gamma_i=-Az_i^2\sqrt II=12∑jcjzj2I=\frac12\sum_j c_jz_j^2

Assumptions. Dilute-solution limiting law; concentrations, standard state and coefficient A must use a consistent convention.

Application. Dilute electrolyte activity corrections.

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 ↗
Vlasov–Poisson model

Couples collisionless distribution dynamics to electrostatic fields.

∂tf+v⋅∇xf−qm∇ϕ⋅∇vf=0\partial_tf+v\cdot\nabla_xf-\frac qm\nabla\phi\cdot\nabla_vf=0−ε0∇2ϕ=ρ-\varepsilon_0\nabla^2\phi=\rho

Assumptions. Multiple species contribute ρ=Σs qs∫fsdv; electrostatic approximation omits electromagnetic induction and radiation.

Application. Electrostatic waves in a plasma.

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

Couples collisionless kinetic distributions to electromagnetic fields.

∂tfs+v⋅∇xfs+qsms(E+v×B)⋅∇vfs=0\partial_tf_s+v\cdot\nabla_xf_s+\frac{q_s}{m_s}(E+v\times B)\cdot\nabla_vf_s=0

Assumptions. Nonrelativistic phase-space form shown; relativistic momentum coordinates are required for high-energy particles.

Application. Kinetic plasma-wave behavior.

Full model, derivation, references, numerical techniques & 3 worked graphs ↗
Magnetohydrodynamics (MHD)

Treats a conducting fluid coupled to a magnetic field.

ρDuDt=−∇p+J×B\rho\frac{Du}{Dt}=-\nabla p+J\times B∂tB=∇×(u×B)+ηm∇2B\partial_tB=\nabla\times(u\times B)+\eta_m\nabla^2B

Assumptions. Simple resistive MHD with constant magnetic diffusivity ηm; continuity, energy and ∇·B=0 complete the system.

Application. Large-scale magnetized-plasma motion.

Full model, derivation, references, numerical techniques & 3 worked graphs ↗
Particle-in-cell (PIC)

Couples moving computational particles to fields on a mesh.

mpv˙p=qp(Ep+vp×Bp)m_p\dot v_p=q_p(E_p+v_p\times B_p)ρgrid=∑pqpW(xgrid−xp)\rho_{\mathrm{grid}}=\sum_pq_pW(x_{\mathrm{grid}}-x_p)

Assumptions. PIC scheme; charge-conserving deposition, field solver, particle shape W and time integration determine numerical behavior.

Application. A kinetic plasma simulation.

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.