Ergodic Properties of Random Billiards Driven by Thermostats

Abstract

Abstract:We consider a class of mechanical particle systems interacting with thermostats. Particles move freely between collisions with disk-shaped thermostats arranged periodically on the torus. Upon collision, an energy exchange occurs, in which a particle exchanges its tangential component of the velocity for a randomly drawn one from the Gaussian distribution with the variance proportional to the temperature of the thermostat. In the case when all temperatures are equal one can write an explicit formula for the stationary distribution. We consider the general case and show that there exists a unique absolutely continuous stationary distribution. Moreover under rather mild conditions on the initial distribution the corresponding Markov dynamics converges to the equilibrium with exponential rate. One of the main technical difficulties is related to a possible overheating of moving particle. However as we show in the paper non-compactness of the particle velocity can be effectively controlled.

Sub-exponential mixing of random billiards driven by thermostats

“…Non-equilibrium steady states for open mechanical particle systems have been extensively studied in recent years as a part of the efforts to derive the Fourier Law from the laws of microscopic dynamics [1, 3, 5, 7, 11-13, 15, 18]. The existence of non-equilibrium steady states has been rigorously shown for only a handful of systems, mostly by using either spectraltheoretic arguments [4,6,8] or Harris' ergodic theorem [12,17]; these methods automatically yield exponential mixing and exponential rates of convergence of initial distributions to the unique steady state. For systems that do not mix exponentially fast, which is frequently the case for continuous-time mechanical particle systems driven by heat reservoirs, general methods to prove the existence of non-equilibrium steady states are virtually non-existent.…”

“…An excellent review of the ideas and difficulties associated with this task is presented in [9]. This paper, together with [12], provides a rigorous proof of the existence of a unique steady state with sub-exponential mixing rates for a class of random billiards driven by thermostats which randomize the velocities only in one direction, leaving a large memory trace in the system. In addition, for a large class of initial distributions, we obtain polynomial lower bounds on the rates of convergence to the steady state.…”

“…Therefore, one can define a discrete Markov chain in position and normal velocity variables that represents the original dynamics at collisions. For this Markov chain, the existence and uniqueness of the invariant probability measure, which describes the steady state, along with exponential rates of convergence of the initial distributions to the invariant measure were obtained in [12] via Harris' ergodic theorem.…”

Entropy production in random billiards

“…In [5], it is shown that the Maxwell-Boltzmann distribution is the equilibrium distribution for a large class of examples with a single heat bath. Ergodicity was proved for reflection models with multiple temperatures in [18], but this is for a special class of billiard tables, using the geometry of dispersing Sinai billiards.…”

Polynomial convergence to equilibrium for a system of interacting particles

“…In this paper, we consider a class of particle systems for which such modeling is more straightforward, namely when the particles do not interact with one another directly but only via their "environment", or a "hub". Concrete examples of mechanical models of this type were introduced in [33,36] and studied later in [10,30,29,11,12,13,44,25]. In these models, the "environment" is symbolized by the kinetic energy stored in rotating disks placed at various locations in the physical domain.…”