Improved Approximation of Infinite Thermostat by Finite Reservoir Using the 3D Kac Model

TL;DR

This paper extends the analysis of particle-reservoir interactions from a 1D model to a 3D Kac Model, establishing a uniform-in-time bound on the difference between finite and infinite thermostat systems, with the bound scaling as M/N.

Contribution

It introduces a 3D Kac Model framework to improve bounds on the approximation of infinite thermostats by finite reservoirs, generalizing previous 1D results.

Findings

Bound scales with M/N over time

Uses 3D Kac Model for better approximation

Establishes uniform-in-time distance bounds

Abstract

In this paper, we study a system of $M$ particles interacting with a reservoir of $N$ particles, where $N >> M$, and compare this setup to one where the $M$-particle system interacts with a thermostat of infinite particles. Our goal is to prove a suitable upper bound, uniform in time, on the distance between the states of these two setups, given an initial Maxwellian state for both the reservoir and thermostat. Previous work has analyzed this problem using the one-dimensional Kac Model of gas collisions and an $L^2$ norm to define distance; the result was a bound which scaled with $M/\sqrt{N}$. In this paper, we use the $L^2$ norm and the three-dimensional generalization of the Kac Model to prove a bound whose long-term behavior scales with $M/N$.