DRIFT AND DIFFUSION FOR A MECHANICAL SYSTEM

We consider a mechanical system in the plane, consisting of a vertical rod of length l, with its center moving on the horizontal axis, subject to elastic collisions with the particles of a free gas, and to a constant force f. Assuming a suitable initial measure we show that the evolution of the system as seen from the rod is described by an exponentially ergodic irreducible Harris chain, implying convergence to a stationary invariant measure as t --> infinity. We deduce that in the proper scaling the motion of the rod is described as a drift plus a diffusion. We prove in conclusion that the diffusion is nondegenerate and that the drift is nonzero if f not equal 0 and has the same sign of f.