Some ideas about quantitative convergence of collision models to their mean field limit

TL;DR

This paper proves the convergence of a stochastic particle system to the Boltzmann equation in a homogeneous setting, providing quantitative bounds on the convergence rate with Gaussian tail estimates.

Contribution

It introduces a quantitative convergence analysis for collision models to their mean field limit, with explicit probabilistic bounds in Sobolev spaces, applicable to Maxwellian models.

Findings

Convergence of particle system to Boltzmann equation as N→∞

Gaussian tail bounds for the empirical measure distance

Numerical results support practical relevance

Abstract

We consider a stochastic $N$-particle model for the spatially homogeneous Boltzmann evolution and prove its convergence to the associated Boltzmann equation when $N\to \infty$. For any time $T>0$ we bound the distance between the empirical measure of the particle system and the measure given by the Boltzmann evolution in some homogeneous negative Sobolev space. The control we get is Gaussian, i.e. we prove that the distance is bigger than $x N^{-1/2}$ with a probability of type $O(e^{-x^2})$. The two main ingredients are first a control of fluctuations due to the discrete nature of collisions, secondly a Lipschitz continuity for the Boltzmann collision kernel. The latter condition, in our present setting, is only satisfied for Maxwellian models. Numerical computations tend to show that our results are useful in practice.