Ergodic Properties of Random Billiards Driven by Thermostats

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.