Rate of convergence of the Nanbu particle system for hard potentials

TL;DR

This paper analyzes the Nanbu stochastic particle system for hard potentials in the Boltzmann equation, establishing a near-optimal convergence rate of the empirical measure to the true solution, with implications for numerical simulations.

Contribution

It provides the first quantitative convergence rate for the Nanbu particle system for hard potentials, improving understanding of its accuracy and efficiency.

Findings

Convergence rate is almost optimal in the number of particles.

The Wasserstein distance between empirical and true measures decreases at a quantifiable rate.

The rate is not uniform over time, indicating temporal limitations.

Abstract

We consider the (numerically motivated) Nanbu 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. More precisely, we estimate the expectation of the squared Wasserstein distance with quadratic cost between the empirical measure of the particle system and the solution. The rate we obtain is almost optimal as a function of the number of particles but is not uniform in time.