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

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 $M/N$.