H-theorem

Statement

Let $\mathbb{T}^3$ be the three-dimensional torus. Let $B \colon [0, \infty) \times [-1, 1] \to [0, \infty)$ be a collision kernel with $B > 0$ almost everywhere. For $v, v_* \in \R^3$ and $\sigma \in S^2$ put $v' = \tfrac12 (v + v_*) + \tfrac12 \abs{v - v_*}\, \sigma$ and $v_*' = \tfrac12 (v + v_*) - \tfrac12 \abs{v - v_*}\, \sigma$. Let $f(t, x, v) > 0$ be a smooth solution on $[0, T) \times \mathbb{T}^3 \times \R^3$ of the Boltzmann equation $$\partial_t f + v \cdot \nabla_x f = \int_{\R^3} \int_{S^2} B\big( \abs{v - v_*}, \cos\theta \big) \big( f' f_*' - f f_* \big) \, d\sigma \, dv_*,$$ where $f' = f(t, x, v')$, $f_*' = f(t, x, v_*')$, $f_* = f(t, x, v_*)$, and $\cos\theta = \sigma \cdot (v - v_*) / \abs{v - v_*}$. Assume that $f$ decays fast enough in $v$ for all the integrals below to converge and for differentiation under the integral sign. Let $H(t) = \int_{\mathbb{T}^3} \int_{\R^3} f \log f \, dv \, dx$. Then $$\frac{dH}{dt} = - \frac14 \int_{\mathbb{T}^3} \int_{\R^3} \int_{\R^3} \int_{S^2} B \, \big( f' f_*' - f f_* \big) \log \frac{f' f_*'}{f f_*} \, d\sigma \, dv_* \, dv \, dx \le 0,$$ and $dH/dt = 0$ at time $t$ if and only if, for every $x$, the function $v \mapsto f(t, x, v)$ is a Maxwell–Boltzmann distribution.