Fluids, Plasmas, and Kinetic Theory

Matter can be described at several resolutions. Continuum theory follows density, velocity, energy, and stress; kinetic theory follows a distribution on phase space; plasma theory retains the relative motion of charged species; and magnetohydrodynamics (MHD) compresses a conducting plasma into one electromagnetic fluid. None is automatically more fundamental for a given calculation. The useful description is the least detailed one whose state variables, scale ordering, closure, and admissibility rule still resolve the question being asked.

Helpful background. Calculus and Linear Algebra for Relativity supplies vector, tensor, eigenvalue, and integration tools; Classical and Variational Mechanics supplies force, work, and conservation laws; States, Processes, and the Laws of Thermodynamics supplies state variables, equations of state, temperature, and entropy; Classical Fields and Their Actions supplies Maxwell theory and field energy; Differential Equations for Relativity supplies transport, linearization, and characteristic reasoning.

Choosing a description by scale and closure

Let $L$ and $t_{\rm mac}$ be the spatial and temporal scales of the phenomenon. A collisional hydrodynamic reduction is controlled by ratios such as

$$\mathrm{Kn}=\frac{\ell_{\rm mfp}}{L}\ll1, \qquad \frac{\tau_{\rm coll}}{t_{\rm mac}}\ll1,$$

where $\ell_{\rm mfp}$ and $\tau_{\rm coll}$ are a mean free path and collision time. These inequalities allow a distribution close to local equilibrium to be represented by a few moments plus a gradient expansion. They do not by themselves choose an equation of state, viscosity, conductivity, or boundary condition.

For a plasma, further ratios decide which charged-species effects survive:

$$\frac{\lambda_D}{L}, \qquad \frac{d_i}{L}, \qquad \frac{\rho_i}{L}, \qquad \frac{\omega}{\Omega_{ci}}, \qquad \frac{U}{c},\quad\frac{c_s}{c},\quad\frac{v_A}{c}.$$

Here $\lambda_D$ is the Debye screening length, $d_i$ the ion inertial length, $\rho_i$ the ion gyroradius, and $\Omega_{ci}$ the ion cyclotron frequency; $U$, $c_s$, and $v_A$ are bulk, sound, and Alfvén speeds. Small Debye length supports bulk quasineutrality; small ion inertial length and gyroradius support a one-fluid reduction. Omitting displacement current requires the nonrelativistic electromagnetic ordering as well as slow evolution. A small bulk speed $U/c$ alone does not ensure that the wave speeds $c_s$ and $v_A$ are small compared with $c$. A thin sheath, shock, current sheet, or resonant particle population can violate the orderings locally even when the global system is large.

The second independent choice is closure. Conservation gives evolution equations for mass, momentum, and total energy, but momentum contains stress and energy contains heat flux. A continuum theory must specify those quantities. A kinetic theory evolves the distribution instead, but must specify its force and collision operator. MHD additionally needs an Ohm law and thermodynamic closure. This distinction between exact balance and model input is the organizing principle of the chapter.

A map of the models

Description	State retained	Governing content	Additional closure or ordering	Typical failure signal
Ideal fluid	$\rho,\mathbf u,e$	Mass, momentum, and energy balance	Equation of state; isotropic pressure; negligible dissipative fluxes	Mean-free-path effects, anisotropic stress, heat flow, or entropy production matters
Dissipative fluid	Ideal variables plus viscous stress and heat flux	Navier–Stokes–Fourier balance and entropy production	Near-local equilibrium; transport coefficients; gradient expansion	Large Knudsen number, rapid transients, or unchecked causality and stability in relativistic closure, especially for traditional parabolic first-order forms
Boltzmann gas	$f(\mathbf x,\mathbf p,t)$	Streaming plus binary collisions	Diluteness, molecular chaos, cross section, boundary data	Strong correlations, quantum degeneracy, or long-range collective fields
Vlasov–Maxwell plasma	Species distributions and electromagnetic fields	Collisionless phase-space transport plus Maxwell equations	Mean-field treatment; resolved velocity space	Collisional relaxation or unresolved small scales control the observable
Single-fluid MHD	$\rho,\mathbf u,p,\mathbf B$	Fluid balances, induction, and magnetic stress	Quasineutral low-frequency large-scale ordering; Ohm law; pressure closure	Sheaths, inertial or gyroradius scales, pressure anisotropy, reconnection microphysics, or relativistic speeds
Weak shock solution	Piecewise-smooth conserved variables	Integral conservation and Rankine–Hugoniot jumps	Entropy or regularization-based admissibility	The chosen regularization selects a different wave family or internal structure

The rows overlap. For example, ideal MHD is simultaneously a continuum closure, a low-frequency electromagnetic reduction, and the zero-nonideal-term limit of a generalized Ohm law. “Use MHD” is therefore not one assumption but a bundle that must be checked against the scales of the requested observable.

Two routes through the chapter

The continuum route starts with Continuum Kinematics and Balance Laws. It distinguishes Eulerian fields from material parcels, localizes moving-volume balances, and derives mass, momentum, energy, and entropy statements; Mei 2002, §1.3, Eqs. (1.3.1)–(1.3.10), printed pp. 1–4 derives the moving-volume identity and localization used there. Ideal Fluids, Vorticity, and Bernoulli Relations adds an equation of state and isotropic pressure, then separates Kelvin, Bernoulli, and Crocco results by their hypotheses. Viscosity, Heat Flow, and Dissipation adds Newtonian stress and Fourier heat flow and checks their entropy production. Continue to Fluid Waves, Shocks, and Instabilities for characteristics, weak solutions, and linear stability.

The microscopic and plasma route starts with Distribution Functions and the Boltzmann Equation. Collision invariants produce the fluid balances, a Maxwellian gives ideal closure, the H-theorem fixes the relaxation direction, and Vlasov–Maxwell theory separates collisionless collective motion from binary collisions. Plasmas and Magnetohydrodynamics then derives total momentum and generalized Ohm law from species equations before taking ideal, resistive, or Hall limits. Finish again with the waves page, where characteristic families and shock admissibility depend on the chosen continuum model.

A first reading can follow the six pages in order. A reader repairing a particular prerequisite can instead enter at the relevant model, but should return to the continuum balance page whenever a sign, flux, or conservation statement is unclear.

Cross-model diagnostic

Suppose a conducting gas has

$$L=10^4\,\mathrm m, \quad \ell_{\rm mfp}=10\,\mathrm m, \quad \lambda_D=10^{-2}\,\mathrm m, \quad d_i=10^2\,\mathrm m,$$ $$\frac{\omega}{\Omega_{ci}}=2\times10^{-2}, \qquad \frac{U}{c}=5\times10^{-2}, \qquad \frac{\tau_{\rm coll}}{t_{\rm mac}}=2\times10^{-3}, \qquad R_m=10^5.$$

The corresponding scale ratios are

$$\mathrm{Kn}=10^{-3}, \qquad \frac{\lambda_D}{L}=10^{-6}, \qquad \frac{d_i}{L}=10^{-2}.$$

For a bulk sound wave on scale $L$, the small spatial and temporal collisional ratios make a fluid description plausible. For the large-scale magnetic evolution, the quoted ratios support several MHD orderings, and the large magnetic Reynolds number $R_m$ supports negligible resistive diffusion away from sharp gradients. The diagnostic is incomplete until ion-gyroradius effects, characteristic speeds $c_s/c$ and $v_A/c$, pressure closure, and the remaining nonideal Ohm-law terms have also been checked. Three different questions further change the answer:

• attenuation of the sound wave requires dissipative transport coefficients rather than ideal Euler flow;
• a current sheet of thickness $50\,\mathrm m\lt d_i$ does not satisfy the one-fluid ion-inertial ordering, even though $d_i/L\ll1$ globally;
• an energetic non-Maxwellian particle tail is a kinetic observable and cannot be recovered from $\rho$, $p$, and $\mathbf u$ alone.

Model choice is therefore question-local and scale-local. A successful global ordering does not authorize the same closure in every subregion or for every observable.

Conserved moments connect the descriptions

The chapter’s models are joined by moment maps. In nonrelativistic kinetic theory,

$$n=\int f\,d^3p, \qquad n\mathbf u=\int\frac{\mathbf p}{m}f\,d^3p,$$

while second and third peculiar-velocity moments give pressure and heat flux. If the collision operator conserves $1$, $\mathbf p$, and $p^2/(2m)$, its corresponding moments vanish, leaving number, momentum, and energy balances. Tong derives this hierarchy in Tong 2012, §§2.2–2.4, printed pp. 24–42.

The moment map loses information. Many distributions share the same density, velocity, and temperature, so those fields do not determine a collisionless tail, anisotropic pressure, or heat flux. A Maxwellian supplies one closure; a Chapman–Enskog correction supplies another; a pressure tensor or higher-moment system retains more information. Every closure should say what information it discards and which scale separation controls the loss.

The same logic connects two-fluid plasma equations to MHD. Summing species momenta preserves total momentum and Lorentz force; subtracting appropriately yields a generalized Ohm law. Ideal MHD then drops resistive, Hall, electron-pressure, and electron-inertia terms only after comparing each with the retained term. Hutchinson develops this species-to-fluid hierarchy in Hutchinson 2006, §4.6, printed pp. 74–77.

From nonrelativistic to relativistic matter

This chapter uses nonrelativistic equations to explain conservation, closure, and scale separation. The following chapters extend these ideas to relativistic matter:

• Relativistic Fluids and Kinetic Theory replaces Galilean densities and fluxes by particle currents and stress–energy, and treats causal dissipative and kinetic matter.
• Relativistic Magnetohydrodynamics and Plasmas combines covariant Maxwell and matter equations and distinguishes ideal, resistive, force-free, and kinetic regimes.
• Relativistic Hydrodynamics, GRMHD, and Radiation turns those covariant systems into conservative numerical variables, primitive recovery, radiation closures, and benchmarks.

The present chapter does not derive relativistic stellar structure, curved-spacetime transport, reconnection microphysics, radiation transfer, or production algorithms. It prepares the diagnostic questions those subjects require: What is conserved? Which variables are evolved? Which closure is used? What scale ordering justifies it? How are discontinuities selected? Where does the model fail?

Review the chapter

A dilute neutral gas has $\mathrm{Kn}\ll1$ in the bulk, but its wall-adjacent layer has thickness comparable to $\ell_{\rm mfp}$. A measured quantity is the force on the wall. Why is a bulk Navier–Stokes solution alone insufficient, even if it accurately predicts the velocity far from the wall?

Solution

The wall force is determined by momentum transfer precisely in the region where the distribution is distorted by the boundary. There the spatial scale is comparable to $\ell_{\rm mfp}$, so the local Knudsen number is order unity and the near-equilibrium gradient closure is not controlled. A bulk continuum solution can still supply outer data, but it must be matched to a kinetic boundary layer or to a boundary condition derived from one. Global small $\mathrm{Kn}$ does not erase a locally kinetic observable.

References

• Hutchinson, Ian H. Introduction to Plasma Physics I, Chapter 4: “Fluid Description of Plasma.” MIT 22.611J course notes (2006 course copy), §4.6, printed pp. 74–77. MIT OpenCourseWare PDF.
• Mei, Chiang C. “Kinematic Transport Theorem.” MIT 1.63J/2.21J Advanced Fluid Dynamics of the Environment notes (2002), §1.3, Eqs. (1.3.1)–(1.3.10), printed pp. 1–4. MIT OpenCourseWare PDF.
• Tong, David. Lectures on Kinetic Theory. University of Cambridge course notes (2012), §§2.2–2.4, printed pp. 24–42. Official author PDF.