Distribution Functions and the Boltzmann Equation

A fluid records only a few moments of microscopic motion. Kinetic theory retains a one-particle distribution, exposes the assumptions behind those moments, and explains why conservation does not by itself close their evolution. Boltzmann and Vlasov equations describe two different mechanisms: short-range collisional relaxation and collisionless self-consistent mean-field motion.

Required background. Classical and Variational Mechanics supplies Hamiltonian phase-space motion, force, momentum, and energy; States, Processes, and the Laws of Thermodynamics supplies equilibrium, entropy, temperature, and irreversibility. Helpful background. Calculus and Linear Algebra for Relativity supplies multidimensional integration and tensor moments; Classical Fields and Their Actions supplies Maxwell fields and conserved currents; Differential Equations for Relativity supplies transport equations and relaxation modes.

The one-particle distribution and free streaming

Use the nonrelativistic convention

$$dN=f(\mathbf x,\mathbf p,t)\,d^3x\,d^3p, \qquad n(\mathbf x,t)=\int f\,d^3p.$$

Then $f$ is number per phase-space volume and is dimensionful. Spin degeneracy or factors of $(2\pi\hbar)^{-3}$ can instead be placed in the measure, but every moment and equilibrium normalization must use the same convention.

With $\mathbf v=\mathbf p/m$ and a force $\mathbf F$ satisfying $\nabla_{\mathbf p}\cdot\mathbf F=0$, the Boltzmann equation is

$$\boxed{ \left( \partial_t+\mathbf v\cdot\nabla_{\mathbf x} +\mathbf F\cdot\nabla_{\mathbf p} \right)f=C[f]. }$$

The left side follows collisionless phase-space trajectories. The stated force condition holds for a position-dependent conservative force and for the Lorentz force; it makes the streaming flow preserve phase-space volume. A nonzero right side represents encounters unresolved by the one-particle trajectory. For a compressible phase-space flow, such as prescribed drag $\mathbf F=-\alpha\mathbf p$, number conservation instead requires the conservative form

$$\partial_t f+\nabla_{\mathbf x}\cdot(\mathbf v f) +\nabla_{\mathbf p}\cdot(\mathbf F f)=C[f].$$

Expanding it adds $f\,\nabla_{\mathbf p}\cdot\mathbf F$ to the streaming equation. Omitting that term would change particle number rather than merely slow the particles.

For a dilute classical gas of identical particles with elastic binary collisions, a standard form is

$$C[f](\mathbf p)= \int d^3p_1\,d\Omega\, g\frac{d\sigma}{d\Omega} \left(f'f_1'-ff_1\right),$$

where $g=\lvert\mathbf v-\mathbf v_1\rvert$, unprimed momenta enter a collision, and primed momenta leave it. The differential cross section and measure are understood to respect time-reversal or microscopic reversibility. The product $ff_1$ is molecular chaos: incoming two-particle correlations are replaced by a product of one-particle distributions. This is the irreversible statistical closure, not a consequence of reversible $N$-body mechanics. Tong 2012, §§2.1–2.2, especially Eqs. (2.6)–(2.13) and (2.23)–(2.24), printed pp. 14–26 develops the BBGKY-to-Boltzmann reduction and its assumptions.

Collision invariants and the complete first moments

Elastic number-conserving collisions satisfy

$$\int d^3p\,\psi(\mathbf p)C[f]=0$$

for the collision invariants

$$\psi=1, \qquad \psi=p_i, \qquad \psi=\frac{p^2}{2m}.$$

Define mass density, bulk velocity, peculiar velocity, pressure tensor, internal-energy density, and heat flux by

$$\rho=mn, \qquad n\mathbf u=\int\mathbf v f\,d^3p, \qquad \mathbf c=\mathbf v-\mathbf u,$$ $$P_{ij}=m\int c_i c_j f\,d^3p, \qquad \varepsilon=\frac{m}{2}\int c^2f\,d^3p, \qquad q_i=\frac{m}{2}\int c^2c_i f\,d^3p.$$

For the following compact force terms, assume $\mathbf F=m\mathbf a(\mathbf x,t)$ is independent of momentum. Multiplying the kinetic equation by each invariant and integrating over momentum gives

$$\boxed{ \partial_tn+\nabla\cdot(n\mathbf u)=0, }$$ $$\boxed{ \partial_t(\rho u_i) +\partial_j(\rho u_i u_j+P_{ij}) =\rho a_i, }$$

and

$$\boxed{ \partial_t\mathcal E +\nabla\cdot \left(\mathcal E\mathbf u+\mathbf P\cdot\mathbf u+\mathbf q\right) =\rho\mathbf a\cdot\mathbf u, }$$

with

$$\mathcal E=\frac12\rho u^2+\varepsilon.$$

Collisions redistribute momentum and energy among particles but do not appear as sources in these three local balances. Tong 2012, §2.4.1, Eqs. (2.45)–(2.56), printed pp. 36–42 derives the hierarchy.

The hierarchy is not closed: momentum evolution contains $\mathbf P$, pressure evolution contains a third moment, and so on. Conservation explains which low moments survive collisions; it does not express higher moments in terms of lower ones. To compare signs with the continuum page, the kinetic pressure tensor is $\mathbf P=-\boldsymbol\sigma$: isotropic equilibrium has $\mathbf P=p\mathbf I$, while a Newtonian viscous closure has $\mathbf P=p\mathbf I-\boldsymbol\tau$.

Maxwellian closure and detailed balance

In local equilibrium,

$$f_{\rm M}= n\left(\frac{1}{2\pi mT}\right)^{3/2} \exp\left[-\frac{(\mathbf p-m\mathbf u)^2}{2mT}\right],$$

where $k_{\mathrm B}=1$. Direct Gaussian integration gives

$$P_{ij}=nT\delta_{ij}, \qquad \varepsilon=\frac32nT, \qquad \mathbf q=0.$$

This is ideal-fluid closure. It also annihilates the whole collision operator, not only its first moments. Elastic conservation gives

$$\mathbf p+\mathbf p_1=\mathbf p'+\mathbf p_1', \qquad E+E_1=E'+E_1'.$$

Therefore

$$\frac{f_{\rm M}'f_{{\rm M}1}'} {f_{\rm M}f_{{\rm M}1}} =\exp\left[-\frac{(E'+E_1')-(E+E_1)}{T}\right]=1,$$

so gain and loss cancel pairwise. This establishes $C[f_{\rm M}]=0$ at each event; it does not make an arbitrary spatially varying Maxwellian an exact solution of the streaming equation. Gradients generally create departures from local equilibrium. Controlled small-gradient corrections to $f_{\rm M}$ generate the viscous stress and heat flux of the previous page. The expansion requires the mean free path and collision time to be small relative to macroscopic scales.

The H-theorem with a dimensionless logarithm

Choose a fixed reference phase-space density $f_\star$ with the same units as $f$ and define

$$H[f]=\int d^3x\,d^3p\, f\ln\left(\frac{f}{f_\star}\right).$$

The reference changes $H$ by a constant multiple of total particle number, so it does not affect $dH/dt$ in a closed system. Assume periodic boundaries or sufficient decay, phase-space-volume-preserving streaming, elastic binary collisions, molecular chaos, and microreversibility. Streaming then contributes only a boundary term; the prescribed drag example above does not satisfy this hypothesis. Symmetrizing incoming particles, outgoing particles, and the reverse collision gives

$$\boxed{ \frac{dH}{dt} =-\frac14\int d\Gamma\,W (x-y)\ln\left(\frac{x}{y}\right)\leq0, }$$

where

$$x=ff_1, \qquad y=f'f_1',$$

and $d\Gamma\,W$ denotes the nonnegative collision measure including $g\,d\sigma/d\Omega$. The sign follows from

$$(x-y)\ln(x/y)\geq0$$

for positive $x$ and $y$. Equality requires detailed balance on every allowed collision. Then $\ln f$ is a linear combination of the collision invariants, which yields a Maxwellian. Tong gives the symmetrization in Tong 2012, §2.3, Eqs. (2.38)–(2.42), printed pp. 33–35 (PDF).

Physical entropy is $S=-H$ up to constants for this classical dilute-gas convention, so it does not decrease. The theorem does not apply unchanged to an arbitrarily correlated $N$-body distribution, dense matter, quantum statistics, or a collision law without the stated reversibility. Its arrow of time enters through the one-particle closure and boundary conditions.

Collisionless plasmas: Vlasov–Maxwell rather than “Boltzmann with zero collisions”

For plasma species $s$ with charge $q_s$ and mass $m_s$, the collisionless mean-field equation is

$$\boxed{ \partial_tf_s +\mathbf v\cdot\nabla_{\mathbf x}f_s +q_s(\mathbf E+\mathbf v\times\mathbf B) \cdot\nabla_{\mathbf p}f_s=0. }$$

The fields are not prescribed independently. They are sourced self-consistently by

$$\rho_q=\sum_s q_s\int f_s\,d^3p, \qquad \mathbf J=\sum_s q_s\int\mathbf v f_s\,d^3p,$$

together with Maxwell’s equations. This is Vlasov–Maxwell theory. It is more than deleting $C[f]$ from an externally forced gas because every particle changes the collective fields that drive every other particle.

The Lorentz force is divergence free in phase space:

$$\nabla_{\mathbf x}\cdot\mathbf v +\nabla_{\mathbf p}\cdot \left[q_s(\mathbf E+\mathbf v\times\mathbf B)\right]=0.$$

Hence Vlasov flow preserves phase-space volume and every fine-grained functional $\int\Phi(f_s)$ under suitable boundaries. Collisionless phase mixing can nevertheless create structure on progressively finer velocity scales, so a coarse-grained observer may see apparent relaxation without Boltzmann entropy production. Hutchinson separates single-particle, kinetic, fluid, and self-consistent field descriptions in Hutchinson 2006, §§1.6–1.6.2, printed pp. 15–16.

Model	Evolved information	Microscopic interaction	Entropy statement	Typical closure boundary
Boltzmann	One-particle $f$	Short-range binary collisions	$H$ decreases under molecular chaos and detailed balance	Dilute gas; collision model specified
Vlasov–Maxwell	One $f_s$ per species plus self-consistent fields	Collisionless collective mean field	Fine-grained functionals are conserved	Scales where binary collisions are negligible
Fluid	Density, velocity, energy, and selected extra moments	Encoded in pressure and transport laws	Entropy production belongs to the chosen closure	Small unresolved kinetic corrections

Worked example: a conservative relaxation model

For a spatially homogeneous gas, consider the Bhatnagar–Gross–Krook (BGK) model

$$\partial_tf=-\frac{f-f_{\rm eq}}{\tau}.$$

If $f_{\rm eq}$ is chosen at each time with the same number, momentum, and energy moments as $f$, then

$$\int d^3p\, \begin{Bmatrix} 1\\[2pt] \mathbf p\\[2pt] p^2/(2m) \end{Bmatrix} (f-f_{\rm eq})=0.$$

The model therefore preserves those moments exactly while relaxing all unmatched components. If $f_{\rm eq}$ is fixed and moment matched to the initial state,

$$f(t)=f_{\rm eq}+[f(0)-f_{\rm eq}]e^{-t/\tau}.$$

Choosing the wrong mean velocity or temperature would spuriously destroy momentum or energy. Moment matching is part of the collision model, not bookkeeping.

Optional bridge: relativistic phase space

The future mass shell is the set of future-directed four-momenta satisfying $p^\mu p_\mu=-m^2$ and $p^0\gt0$; $\Gamma^i_{\alpha\beta}$ are the spacetime metric’s Levi-Civita connection coefficients in the chosen coordinates. Readers new to these objects can review Events, Intervals, and Minkowski Geometry and The Levi-Civita Connection.

Set $c=\hbar=1$ and let $g_{\rm int}$ count internal states, such as spin polarizations. To display a common relativistic normalization, use a new scalar distribution $f_{\rm rel}$ with the internal-state and phase-space factors in the measure. On the future mass shell,

$$d\Pi=\frac{g_{\rm int}\,d^3p}{(2\pi)^3p^0}, \qquad N^\mu=\int d\Pi\,p^\mu f_{\rm rel}, \qquad T^{\mu\nu}=\int d\Pi\,p^\mu p^\nu f_{\rm rel}.$$

On an inertial constant-time slice, $N^0=\int g_{\rm int}f_{\rm rel}\,d^3p/(2\pi)^3$. Agreement with the earlier convention $dN=f\,d^3x\,d^3p$ therefore requires $f=g_{\rm int}f_{\rm rel}/(2\pi)^3$ when both describe the same population. Other conventions place a factor of two in the invariant measure and compensate in the distribution. In curved spacetime, for particles with no external nongravitational force, the scalar distribution obeys

$$p^\mu\partial_\mu f_{\rm rel} -\Gamma^i_{\alpha\beta}p^\alpha p^\beta \frac{\partial f_{\rm rel}}{\partial p^i}=C[f_{\rm rel}].$$

Sarbach and Zannias construct the mass-shell Liouville flow in Sarbach and Zannias 2013, §6, Eqs. (44)–(49), printed pp. 10–13 and derive the moment balances in §§7–9, especially Eqs. (70)–(77), printed pp. 14–24. Collision invariants yield $\nabla_\mu N^\mu=0$ and $\nabla_\mu T^{\mu\nu}=0$ when there are no external exchanges. Full curved-spacetime transport, radiation collision terms, and compact-object numerics belong to later volumes.

Plasmas and Magnetohydrodynamics next asks which scale orderings allow the species distributions and electromagnetic fields to be compressed into one conducting fluid.

Exercises

Vlasov conservation of particle number

Integrate the Vlasov equation over all $\mathbf x$ and $\mathbf p$. Under periodic or decaying boundaries, show that the particle number of each species is conserved.

Solution

Both transport terms are divergences in phase space. The spatial integral of $\nabla_{\mathbf x}\cdot(\mathbf v f_s)$ and the momentum integral of $\nabla_{\mathbf p}\cdot[q_s(\mathbf E+\mathbf v\times\mathbf B)f_s]$ vanish at the boundary. Therefore

$$\frac{d}{dt}\int f_s\,d^3x\,d^3p=0.$$

An anisotropic distribution is not an ideal fluid

Suppose the peculiar-velocity covariance is diagonal with unequal entries. Which fluid variable records the anisotropy, and why can one scalar pressure not close momentum balance?

Solution

The pressure tensor

$$P_{ij}=m\int c_i c_jf\,d^3p$$

has unequal diagonal components. Momentum balance contains $\partial_jP_{ij}$, so replacing $\mathbf P$ by $p\mathbf I$ discards direction-dependent forces. One needs an anisotropic pressure closure or enough kinetic information to evolve the tensor.

References

• Hutchinson, Ian H. Introduction to Plasma Physics I, Chapter 1. MIT 22.611J course notes (2006 course copy), §§1.6–1.6.2, printed pp. 15–16. MIT OpenCourseWare PDF.
• Sarbach, Olivier, and Thomas Zannias. “Relativistic Kinetic Theory: An Introduction.” AIP Conference Proceedings 1548 (2013): 134–155, §§6–9, especially Eqs. (44)–(49) and (70)–(77), manuscript printed pp. 10–24. arXiv:1303.2899; DOI.
• Tong, David. Kinetic Theory. University of Cambridge lecture notes (2012), §§2.1–2.6, especially Eqs. (2.6)–(2.13), (2.23)–(2.24), (2.36)–(2.42), and (2.45)–(2.57), printed pp. 14–52. Official author PDF.