Propagation of Chaos for a Thermostated Kinetic Model

TL;DR

This paper proves that in a system of particles with a Gaussian thermostat and external field, the particle velocity distribution converges to a self-consistent Vlasov-Boltzmann equation as the number of particles grows large, demonstrating propagation of chaos.

Contribution

It establishes propagation of chaos and derives a Vlasov-Boltzmann equation for a thermostated kinetic model with mean-field interactions.

Findings

Large N limit leads to a Vlasov-Boltzmann equation

Propagation of chaos is proven for the model

Thermostat maintains constant kinetic energy

Abstract

We consider a system of N point particles moving on a d-dimensional torus. Each particle is subject to a uniform field E and random speed conserving collisions. This model is a variant of the Drude-Lorentz model of electrical conduction. In order to avoid heating by the external field, the particles also interact with a Gaussian thermostat which keeps the total kinetic energy of the system constant. The thermostat induces a mean-field type of interaction between the particles. Here we prove that, starting from a product measure, in the large N limit, the one particle velocity distribution satisfies a self consistent Vlasov-Boltzmann equation.. This is a consequence of "propagation of chaos", which we also prove for this model.