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

TL;DR

This paper provides a counterexample showing that entropy production can increase over time in the space-homogeneous Boltzmann equation, challenging a long-standing conjecture.

Contribution

It constructs a specific example with a non-physically motivated collision kernel where entropy production increases, disproving McKean's 1966 conjecture.

Findings

Entropy production can increase over time in certain cases.

The collision kernel used is not physically motivated.

The result disproves a longstanding conjecture.

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.