Rate of convergence of the Kac particle system for the Boltzmann equation with hard potentials

TL;DR

This paper establishes an explicit rate of convergence for the Kac particle system to the Boltzmann equation's solution for hard potentials, using a double coupling technique under certain initial data conditions.

Contribution

It provides the first explicit convergence rate for the Kac particle system towards the Boltzmann equation with hard potentials, under finite exponential moment assumptions.

Findings

Explicit convergence rate in Wasserstein distance established.

Convergence proven under finite exponential moment initial data.

Double coupling technique effectively used for analysis.

Abstract

In this paper, we prove that the Kac stochastic particle system converges to the weak solution of the spatially homogeneous Boltzmann equation for hard potentials and hard spheres. We give, under the initial data with finite exponential moment assumption, an explicit rate of propagation of chaos in squared Wasserstein distance with quadratic cost by using a double coupling technique.