A Kac system interacting with two heat reservoirs

TL;DR

This paper analyzes a 3D particle system interacting with two heat reservoirs via a Kac-type model, showing that for short times the reservoirs behave like Maxwellian thermostats, extending previous results to three dimensions.

Contribution

It demonstrates that for short times, the finite reservoirs can be approximated by infinite Maxwellian thermostats, extending prior 2D results to 3D systems.

Findings

Interaction with reservoirs approximates Maxwellian thermostats for short times

Extension of previous 2D results to 3D particle systems

Valid for times much shorter than the square root of reservoir particle number

Abstract

We study a system formed by $M$ particles moving in 3 dimension and interacting with 2 heat reservoirs with $N>>M$ particles each. The system and the reservoirs evolve and interact via random collision described by a Kac-type master equation. The initial state of the reservoirs is given by 2 Maxwellian distributions at temperature $T_+$ and $T_-$. We show that, for times much shorter than $\sqrt{N}$ the interaction with the reservoirs is well approximated by the interaction with 2 Maxwellian thermostats, that is, heat reservoirs with $N=\infty$. As a byproduct, if $T_+=T_-$ we extend the results in \cite{BLTV} to particles in 3 dimension.