  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.
