Partially Thermostated Kac Model

TL;DR

This paper analyzes a partially thermostated Kac model, revealing how the spectral gap and entropy evolve as the number of particles grows, and explores the connection between different thermostats in kinetic theory.

Contribution

It provides new insights into the spectral gap and entropy decay rates in a partially thermostated Kac model, linking it to existing thermostat models via a van Hove limit.

Findings

Spectral gap scales as m/N for large N

Relative entropy approaches equilibrium exponentially at rate ~ m/N^2

Established a relationship between different thermostats in kinetic models

Abstract

We study a system of $N$ particles interacting through the Kac collision, with $m$ of them interacting, in addition, with a Maxwellian thermostat at temperature $\frac{1}{\beta}$. We use two indicators to understand the approach to the equilibrium Gaussian state. We prove that i) the spectral gap of the evolution operator behaves as $\frac{m}{N}$ for large $N$ ii) the relative entropy approaches its equilibrium value (at least) at an eventually exponential rate $\sim \frac{m}{N^2}$ for large $N$. The question of having non-zero entropy production at time $0$ remains open. A relationship between the Maxwellian thermostat and the thermostat used in Bonetto, Loss, Vaidyanathan (J. Stat. Phys. 156(4):647-667, 2014) is established through a van Hove limit.