A consistency estimate for Kac's model of elastic collisions in a dilute gas

TL;DR

This paper provides an explicit Wasserstein distance estimate comparing empirical velocity distributions of two Kac models with different particle counts, demonstrating high-probability agreement and implications for the Boltzmann equation's well-posedness.

Contribution

It introduces a quantitative estimate for the convergence of Kac's model with different particle numbers, linking it to the well-posedness of the Boltzmann equation in measure-valued processes.

Findings

High-probability agreement within $N^{-1/d}$ for large particle numbers

Establishment of the Boltzmann equation as a good approximation

Derivation of a lemma on total variation of time-integrals

Abstract

An explicit estimate is derived for Kac's mean-field model of colliding hard spheres, which compares, in a Wasserstein distance, the empirical velocity distributions for two versions of the model based on different numbers of particles. For suitable initial data, with high probability, the two processes agree to within a tolerance of order $N^{-1/d}$, where $N$ is the smaller particle number and $d$ is the dimension, provided that $d\ge3$. From this estimate we can deduce that the spatially homogeneous Boltzmann equation is well posed in a class of measure-valued processes and provides a good approximation to the Kac process when the number of particles is large. We also prove in an appendix a basic lemma on the total variation of time-integrals of time-dependent signed measures.