Propagation of chaos for the homogeneous Boltzmann equation with moderately soft potentials

TL;DR

This paper proves the propagation of chaos for the homogeneous Boltzmann equation with moderately soft potentials by showing convergence of the Kac particle system to the Boltzmann solution, using Fisher information estimates.

Contribution

It adapts recent techniques to demonstrate convergence in the moderately soft potential regime, controlling interaction singularities via Fisher information.

Findings

Kac particle system converges to the Boltzmann solution as particle number increases

Fisher information is nonincreasing along solutions, aiding control of singularities

Establishes propagation of chaos for the specified potential regime

Abstract

We show that the Kac particle system converges, as the number of particles tends to infinity, to the solution of the homogeneous Boltzmann equation, in the regime of moderately soft potentials, $\gamma \in (-2,0)$ with the common notation. This proves the propagation of chaos. We adapt the recent work of Imbert, Silvestre and Villani, to show that the Fisher information is nonincreasing in time along solutions to the Kac master equation. This estimate allows us to control the singularity of the interaction.