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, subje ct to elastic collisions with the particles of a free gas, and to a co nstant force f. Assuming a suitable initial measure we show that the e volution of the system as seen from the rod is described by an exponen tially ergodic irreducible Harris chain, implying convergence to a sta tionary invariant measure as t --> infinity. We deduce that in the pro per scaling the motion of the rod is described as a drift plus a diffu sion. We prove in conclusion that the diffusion is nondegenerate and t hat the drift is nonzero if f not equal 0 and has the same sign of f.