The entropy production is not always monotone in the space-homogeneous Boltzmann equation

Abstract

We show an example of a function and a collision kernel for which the entropy production increases in time when we flow it by the space-homogeneous Boltzmann equation. The collision kernel is not any of the physically motivated kernels that are commonly used in the literature. In this particular setting, our result disproves a conjecture of McKean from 1966.

Figures (6)

• Figure 1: Configuration where vertices form a square of side length one.
• Figure 2: Level sets of the function $f$. The dark region near the origin is the ball $B_\rho$ where $f = ca^2$. The gray ring at radius $\sqrt{5}$ is where $f = a$. In the rest of $B_{5}$, $f = 1$.
• Figure 3: The first scenario for $Q_+$ happens when $f(v') = f(v'_*) = a$. Here $v' = (2,1)$ and $v'_* = (1,2)$. The four points form a square of side length one, with $|v| = \sqrt 2$ and $|v_*| = 2\sqrt 2$.
• Figure : $Q_+(v) \approx a^2$
• Figure : $Q_+(v) \approx a^2$

Theorems & Definitions (3)

• Theorem 1
• Lemma 2
• Remark 3