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Drift and diffusion for a mechanical system
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  • Published: September 1995

Drift and diffusion for a mechanical system

  • C. Boldrighini1 &
  • M. Soloveitchik2 

Probability Theory and Related Fields volume 103, pages 349–379 (1995)Cite this article

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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References

  1. Boldrighini, C.: Bernoulli property for a one-dimensional system with localized interaction. Comm. Math. Phys.103, 499–514 (1986)

    Google Scholar 

  2. 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)

    Google Scholar 

  3. 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

  4. 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)

    Google Scholar 

  5. 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

    Google Scholar 

  6. 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)

    Google Scholar 

  7. 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)

    Google Scholar 

  8. Erdös, L., Tuyen, D.q.: Central limit theorems for the one-dimensional Rayleigh gas with semipermeable barriers. Comm. Math. Phys.143, 451–466 (1992)

    Google Scholar 

  9. 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

  10. 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)

    Google Scholar 

  11. 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)

    Google Scholar 

  12. Jacod, J., Shiryaev, A.N.: Limit theorems for stochastic processes. Berlin: Springer 1980

    Google Scholar 

  13. Kerstan, J., Matthes, K., Mecke, J.: Infinitely divisible point processes. New York: Wiley 1978

    Google Scholar 

  14. 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)

  15. Nummelin, E.: General irreducible Markov chains and nonnegative operators. Cambridge: Cambridge University Press 1989

    Google Scholar 

  16. Reed, M., Simon, B.: Methods of modern mathematical physics, II, New York: Academic Press 1975

    Google Scholar 

  17. 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)

    Google Scholar 

  18. 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)

    Google Scholar 

  19. Spohn, H.: Large scale dynamics of interacting particles. Berlin: Springer 1991

    Google Scholar 

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Author information

Authors and Affiliations

  1. Dipartimento di Matematica e Fisica, Università di Camerino, Via Madonna delle Carceri, I-62032, Camerino, Italy

    C. Boldrighini

  2. Institut für angewandte Mathematik, Universität Heidelberg, Im Neuenheimer Feld g.294, D-69120, Heidelberg, Germany

    M. Soloveitchik

Authors
  1. C. Boldrighini
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  2. M. Soloveitchik
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Additional information

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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  • Received: 10 November 1994

  • Revised: 15 March 1995

  • Issue Date: September 1995

  • DOI: https://doi.org/10.1007/BF01195479

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Keywords

  • Stochastic Process
  • Probability Theory
  • Horizontal Axis
  • Mechanical System
  • Invariant Measure
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