Rate of convergence of the Kac-like particle system

TL;DR

This paper establishes a convergence rate for a particle system modeling the Boltzmann equation with hard potentials, using probabilistic coupling to quantify how quickly the empirical measure approaches the true solution.

Contribution

It provides the first explicit convergence rate for the Kac-like particle system with true hard potentials, under certain initial conditions.

Findings

Convergence rate of the particle system to the Boltzmann solution is quantified.

Probabilistic coupling method effectively measures propagation of chaos.

Results apply to singular interactions in the Boltzmann framework.

Abstract

In this paper, we consider the Kac stochastic particle system associated to the spatially homogeneous Boltzmann equation for true hard potentials. We establish a rate of propagation of chaos of the particle system to the unique solution of the Boltzmann equation. We use a probabilistic coupling method and give, under suitable assumptions on the initial condition, a rate of convergence of the empirical measure of the particle system to the solution of the Boltzmann equation for this singular interaction.