Approach to Equilibrium in a Microscopic Model of Friction

We consider the time evolution of a disk under the action of a constant force and interacting with a free gas in the mean-field approximation. Letting V₀>0 be the initial velocity of the disk and V_∞>0 its equilibrium velocity, namely the one for which the external field is balanced by the friction force exerted by the background, we show that, if V_∞−V₀ is positive and sufficiently small, then the disk reaches V_∞ with the power law t^−(d+2), d=1,2,3 being the dimension of the physical space. The reason for this behavior is the long tail memory due to recollisions. Any Markovian approximation (or simply neglecting the recollisions) yields an exponential approach to equilibrium.