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Drift and diffusion for a mechanical system. (English) Zbl 0832.60075

Summary: We consider a mechanical system in the plane, consisting of a vertical rod of length \(\ell\), 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 \(t \to \infty\). 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 \(f \neq 0\) and has the same sign of \(f\).

MSC:

60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
60F15 Strong limit theorems
60J60 Diffusion processes
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