Boltzmann H-theorem

Statement (kinetic theory setting)

Let $f(t,x,v)\ge 0$ be the one-particle distribution of a dilute classical gas evolving by the Boltzmann equation

where $Q$ is the binary-collision operator with a nonnegative collision kernel.

Define the Boltzmann H-functional

(typically on a domain with suitable boundary conditions, or in the spatially homogeneous case).

Then, for sufficiently regular solutions and admissible collision kernels,

with equality (for appropriate classes of solutions) if and only if $f$ is a local Maxwellian (i.e., a collision equilibrium).

Equivalently, the kinetic entropy $S_{\mathrm{kin}}(t):= -k_B\,H[f](t)$ is nondecreasing, matching the direction of the second law at this level of description.

Entropy production formula

In the spatially homogeneous case (or after integrating out transport terms), one can write

where $(v,v_1)\mapsto(v',v_1')$ is the post-collision velocity map, $B\ge 0$ is the collision kernel, and the inequality uses $(a-b)\ln(a/b)\ge 0$ for $a,b>0$.

Assumptions and scope

• Dilute gas + binary collisions: encoded in the collision operator $Q(f,f)$.
• Molecular chaos (Stosszahlansatz): the crucial closure assumption leading from reversible microscopic dynamics to the irreversible kinetic equation.
• Regularity / integrability: needed to justify differentiating $H[f]$ and exchanging integrals.

Conceptual meaning

• The H-theorem provides a precise sense in which coarse-grained entropy increases for kinetic evolution, even though the underlying microscopic dynamics (at the microstate level) are reversible.
• It is a prototype mechanism for emergence of irreversibility and motivates connections between kinetic entropy and thermodynamic entropy .