Variational derivation of the homogeneous Boltzmann equation

TL;DR

This paper presents a variational approach to derive the homogeneous Boltzmann equation with hard-sphere interactions, establishing energy conservation and linking solutions to microscopic Kac's walk dynamics.

Contribution

It introduces a variational formulation that uniquely determines energy conserving solutions and connects them to microscopic particle dynamics.

Findings

The variational formulation selects the unique energy conserving solution.

The solution is shown to originate from Kac's walk dynamics.

Propagation of entropic chaoticity over time is established.

Abstract

We introduce a variational formulation of the homogeneous Boltzmann equation, with hard-sphere cross section, which selects the unique energy conserving solution. We prove that this solution arises from the microscopic dynamics, namely Kac's walk, and we establish the propagation in time of entropic chaoticity, under the minimal assumption that the initial distribution is entropically chaotic.