Improved Approximation of Infinite Thermostat by Finite Reservoir Using the 3D Kac Model
Federico Bonetto, Anthony Popa, Matthew Powell, Peter Chen, Steven Tung
TL;DR
This work extends the kinetic-theory analysis of thermostat approximations to a three-dimensional Kac model, comparing an M-particle system coupled to a finite N-particle reservoir with the corresponding thermostat limit. The authors develop an $L^2$-based framework by factoring the state as $f_t = h_t\Gamma$ and introducing a momentum- and energy-preserving rotation average operator $\mathbf{R}$ to relate the finite-reservoir dynamics to the thermostat. They prove a uniform-in-time bound on the distance between the two evolutions, with a short-time $O(M/\sqrt{N})$ contribution and a long-time $O(M/N)$ scaling, achieved via a combination of spectral-gap arguments and a detailed analysis of the averaging operators (Lemmas 1–3) and a Duhamel expansion. The main result quantifies how well a finite reservoir approximates an infinite thermostat for the 3D Kac model, with implications for modeling reservoirs in kinetic systems and for understanding the role of momentum conservation in such approximations.
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 $M/N$.