On a thermostated Kac model with rescaling

TL;DR

This paper introduces a thermostated Kac model with velocity rescaling to maintain energy, proving ergodicity, analyzing equilibrium distributions, and establishing propagation of chaos with explicit convergence rates.

Contribution

It presents a novel global thermostat mechanism for Kac's model, analyzes the resulting equilibrium properties, and proves propagation of chaos with explicit rates.

Findings

Ergodicity of the nonlinear kinetic equation is established.

Equilibrium distributions can have heavy tails and varied regularity.

Propagation of chaos holds with a rate of order N^{-ta} in Wasserstein metric.

Abstract

We introduce a global thermostat on Kac's 1D model for the velocities of particles in a space-homogeneous gas subjected to binary collisions, also interacting with a (local) Maxwellian thermostat. The global thermostat rescales the velocities of all the particles, thus restoring the total energy of the system, which leads to an additional drift term in the corresponding nonlinear kinetic equation. We prove ergodicity for this equation, and show that its equilibrium distribution has a density that, depending on the parameters of the model, can exhibit heavy tails, and whose behaviour at the origin can range from being analytic, to being $C^k$, and even to blowing-up. Finally, we prove propagation of chaos for the associated $N$-particle system, with a uniform-in-time rate of order $N^{-\eta}$ in the squared $2$-Wasserstein metric, for an explicit $\eta \in (0, 1/3]$.