A stochastic particle system approximating the BGK equation

TL;DR

This paper introduces a stochastic particle system that approximates the BGK equation, where thermalization depends only on local particle neighborhoods, improving upon previous models by incorporating local interactions.

Contribution

It demonstrates propagation of chaos and convergence to the BGK equation using a localized thermalization mechanism based on nearby particles.

Findings

Propagation of chaos established.

Convergence to BGK equation proven.

Local neighborhood-based thermalization improves model realism.

Abstract

We consider a stochastic $N$-particle system on a torus in which each particle moving freely can instantaneously thermalize according to the particle configuration at that instant. Following [2], we show that the propagation of chaos does hold and that the one-particle distribution converges to the solution of the BGK equation. The improvement with respect to [2] consists in the fact that here, as suggested by physical considerations, the thermalizing transition is driven only by the restriction of the particle configuration in a small neighborhood of the jumping particle. In other words, the Maxwellian distribution of the outgoing particle is computed via the empirical hydrodynamical fields associated to the fraction of particles sufficiently close to the test particle and not, as in [2], via the whole particle configuration. [2]: arXiv:2002.10535 (Journal reference: Arch. Ration. Mech. Anal. Vol. 240 (2021), pp. 785-808)