Boltzmann H-functional

Definition (dilute classical gas)

Let $f(t,x,v)\ge 0$ be the one-particle distribution on phase space $(x,v)\in \mathbb{R}^d\times\mathbb{R}^d$ for a dilute gas evolving (under molecular chaos) by the Boltzmann equation . The Boltzmann H-functional is

(with the convention $0\ln 0=0$). Often one also uses a spatially homogeneous version $H[f]=\int f(v)\ln f(v)\,dv$.

It is related to an entropy functional by

This $S[f]$ is the kinetic-theory counterpart of thermodynamic entropy and is closely aligned with Shannon entropy / Gibbs–Shannon entropy .

H-theorem (monotonicity)

For sufficiently regular solutions of the Boltzmann equation,

with equality iff $f$ is (locally) Maxwellian. Equivalently, $S[f(t)]$ is nondecreasing.

A standard explicit form of the entropy production is

where $(f,f_1)$ and $(f',f_1')$ denote pre/post-collisional values and $d\Gamma$ abbreviates the collision integration measure; the inequality follows from $(a-b)\ln(a/b)\ge 0$ for $a,b>0$.

This monotonicity is the functional core of the Boltzmann H-theorem .

Equilibrium characterization

In the spatially homogeneous case with fixed mass and energy, the minimizers of $H$ are Maxwell–Boltzmann distributions

where $T$ is the temperature and $u$ is the mean velocity.

Conceptual connections

• The decay of $H$ can be reframed as decrease of a relative entropy to equilibrium (after appropriate normalization), clarifying “entropy production” as a divergence.
• For stochastic dynamics (e.g. master equation models), analogous Lyapunov functionals are built from relative entropy and link to detailed balance .