Pathwise Convergence of the Hard Spheres Kac Process

TL;DR

This paper establishes long-time convergence estimates for the hard spheres Kac process to the Boltzmann equation, demonstrating uniform propagation of chaos with polynomial rates under certain initial conditions.

Contribution

It introduces new Wasserstein-based deviation estimates and a novel continuity estimate for the Boltzmann flow, advancing understanding of long-time particle system behavior.

Findings

Uniform in time propagation of chaos with polynomial rates

New Wasserstein continuity estimate for Boltzmann flow

Deviation bounds for the Kac process from the Boltzmann equation

Abstract

We derive two estimates for the deviation of the $N$-particle, hard-spheres Kac process from the corresponding Boltzmann equation, measured in expected Wasserstein distance. Particular care is paid to the long-time properties of our estimates, exploiting the stability properties of the limiting Boltzmann equation at the level of realisations of the interacting particle system. As a consequence, we obtain an estimate for the propagation of chaos, uniformly in time and with polynomial rates, as soon as the initial data has a $k^\mathrm{th}$ moment, $k>2$. Our approach is similar to Kac's proposal of relating the long-time behaviour of the particle system to that of the limit equation. Along the way, we prove a new estimate for the continuity of the Boltzmann flow measured in Wasserstein distance.