Summary
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 forcef. 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 ast→∞. 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 iff≠0 and has the same sign off.
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Boldrighini, C.: Bernoulli property for a one-dimensional system with localized interaction. Comm. Math. Phys.103, 499–514 (1986)
Boldrighini, C., Cosimi, G., Frigio, S., Nogueira, A.: Convergence to a stationary state and diffusion for a charged particle is a standing medium. Prob. Theory Rel. Fields80, 481–500 (1989)
Boldrighini, C., Dobrushin, R.L., Sukhov, Yu.M.: Time asymptotics for some degenerate models of the evolution of infinite particle systems. Università di Camerino, 1980
Boldrighini, A., Pellegrinotti, A., Presutti, E., Sinai, Ya.G., Soloveychik, M.R.: Ergodic properties of a semi-infinite one-dimensional system of statistical mechanics. Comm. Math. Phys.101, 363–382 (1985)
Calderoni, P., Dürr, D.: The Smoluchovsky limit for a simple mechanical model. In: Calderoni, P.: I suoi lavori, Libreria ed Univ., Tor Vergata, Roma, 1990
Chernov, N.I., Eyink, G.L., Lebowitz, J.L., Sinai, Ya.G.: Steady-state electrical conduction in the periodic Lorentz gas. Comm. Math. Phys.154, 569–601 (1993)
Dürr, D., Goldstein, S., Lebowitz, J.L.: Asymptotics of particle trajectories in infinite one-dimensional systems with collisions. Comm. Pure Appl. Math.38, 573–597 (1985)
Erdös, L., Tuyen, D.q.: Central limit theorems for the one-dimensional Rayleigh gas with semipermeable barriers. Comm. Math. Phys.143, 451–466 (1992)
Ferrari, P., Goldstein, S., Lebowitz, J.L.: In: Fritz, J., Jaffe, A., Szász, D., (eds.) Statistical physics and dynamical systems, pp. 405–439. Birkhäuser, 1985
Goldstein, S., Lebowitz, J.L., Ravishankar, K.: Ergodic properties of a system in contact with a heath bath: a one-dimensional model. Comm. Math. Phys.85, 419–427 (1982)
Goldstein, S., Lebowitz, J.L., Ravishankar, K.: Approach to equilibrium in systems in contact with a heath bath. J. Stat. Phys.43, 303–315 (1986)
Jacod, J., Shiryaev, A.N.: Limit theorems for stochastic processes. Berlin: Springer 1980
Kerstan, J., Matthes, K., Mecke, J.: Infinitely divisible point processes. New York: Wiley 1978
Lebowitz, J.L., Rost, H.: The Einstein relation for the displacement of a test particle in a random environment. Preprint 94-15 (SFB 359), Heidelberg University (1994)
Nummelin, E.: General irreducible Markov chains and nonnegative operators. Cambridge: Cambridge University Press 1989
Reed, M., Simon, B.: Methods of modern mathematical physics, II, New York: Academic Press 1975
Soloveichik, M.R.: Ergodic properties of a system of a statistical mechanics with an external potential (in Russian). Izv. Akad. Nauk SSSR, Ser. Mat.53 (1) 179–199 (1989)
Soloveichik, M.R.: Sufficient condition for bernoulliness of a K system and its application in classical statistical mechanics (in Russian). Mat. Zametki45 (2) 105–112 (1989)
Spohn, H.: Large scale dynamics of interacting particles. Berlin: Springer 1991
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Partially supported by C.N.R. (G.N.F.M.) and M.U.R.S.T. research funds.
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Boldrighini, C., Soloveitchik, M. Drift and diffusion for a mechanical system. Probab. Th. Rel. Fields 103, 349–379 (1995). https://doi.org/10.1007/BF01195479
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DOI: https://doi.org/10.1007/BF01195479