On the Motion of a Body in Thermal Equilibrium Immersed in a Perfect Gas

We consider a body immersed in a perfect gas and moving under the action of a constant force. Body and gas are in thermal equilibrium. We assume a stochastic interaction body/medium: when a particle of the medium hits the body, it is absorbed and immediately re-emitted with a Maxwellian distribution. This system gives rise to a microscopic model of friction. We study the approach of the body velocity V (t) to the limiting velocity V∞ and prove that, under suitable smallness assumptions, the approach to equilibrium is |V (t) − V∞| ≈ C td+1 , where d is the dimension of the space, and C is a positive constant. This approach is not exponential, as typical in friction problems, and even slower than for the same problem with elastic collisions. Mathematics Subject Classification. 76P05, 82B40, 82C40, 35L45, 35L50. Received March 31, 2007. Revised October 16, 2007.