Uniform Approximation of a Maxwellian Thermostat by Finite Reservoirs

Abstract

We study a system of M particles in contact with a large but finite reservoir of N >> M particles within the framework of the Kac master equation modeling random collisions. The reservoir is initially in equilibrium at temperature T = β −1 . We show that for large N, this evolution can be approximated by an effective equation in which the reservoir is described by a Maxwellian thermostat at temperature T . This approximation is proven for a suitable L 2 norm as well as for the Gabetta-Toscani-Wennberg (GTW) distance and is uniform in time.

Entropy Decay for the Kac Evolution

“…Thus, one might guess that if a "small" system of M particles out of equilibrium interacts with a reservoir, that is a large system of N ≥ M particles in thermal equilibrium, then the entropy decays exponentially fast in time. This intuition is also supported by the results in [6]. There it was shown that if the thermostat is replaced by a large but finite reservoir initially in thermal equilibrium, this evolution is close to the evolution given by the thermostat.…”

“…The aim of this section is to rewrite (6), that is e Lt F 0 , in a way which is reminiscent of the Ornstein-Uhlenbeck process. This representation will naturally lead to the next step in the proof of Theorem 2.1, namely the entropy inequality that will be presented in Theorem 4.1.…”

“…In a recent papers [1,2] a different approach is proposed. One considers a small system with M particles in contact with a large heat reservoir with N particles.…”

The BGK Equation as the Limit of an N-Particle System

“…The objective of this paper is to prove that when N , M → ∞ the density of one passive particle, f (10) N M (v, t) converges to a function f (v, t) that satisfies the spatially homogeneous BGK equation (5). This is formulated in the following theorem:…”

“…The paper is based on the results in the doctoral thesis of the first author [19]. A very similar model, with two different kinds of particles, has been presented by Bonetto et al in [6], and also in [5]. The authors are in general interested in kinetic models coupled with a thermostat, and in the cited papers the larger set of particles (N in our paper) is considered as a thermostat acting on the smaller set of particles, and they prove that indeed, when N → ∞ the large N -particle system is a good approximation of a Gaussian thermostat.…”