Probability Theory and Related Fields

, Volume 99, Issue 2, pp 211–236 | Cite as

Stationary states of random Hamiltonian systems

  • J. Fritz
  • T. Funaki
  • J. L. Lebowitz
Article
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Revised:

Summary

We investigate the ergodic properties of Hamiltonian systems subjected to local random, energy conserving perturbations. We prove for some cases, e.g. anharmonic crystals with random nearest neighbor exchanges (or independent random reflections) of velocities, that all translation invariant stationary states with finite entropy per unit volume are microcanonical Gibbs states. The results can be utilized in proving hydrodynamic behavior of such systems.

Mathematics Subject Classification (1991)

60K35 82A05 
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References

  1. [A] Arnold, V.I.: Mathematical methods of classical mechanics. Moscow: Hayka 1974Google Scholar
  2. [DS] Deuschel, J.D., Stroock, D.W.: Large deviations. New York: Academic Press 1989Google Scholar
  3. [DV] Donsker, M.D., Varadhan, S.R.S.: Asymptotic evaluation of certain Markov process expectations for large time I. Commun. Pure. Appl. Math.28, 1–47 (1975)Google Scholar
  4. [E] Echeverria, P.: A criterion for invariant measures of Markov processes. Z. Wahrscheinlechkeitstheor. Verw. Geb.61, 1–16 (1982)Google Scholar
  5. [ES] Eyink, G., Spohn, H.: Space-time invariant states of the ideal gas with finite number, energy and entropy density. (Preprint, 1991)Google Scholar
  6. [Fr1] Fritz, J.: An ergodic theorem for the stochastic dynamics of quasi-harmonic crystals. In: Fritz, J., Lebowitz, J.L., Szász, D. (eds.) Random fields. Proceedings of the colloquium on random fields. Esztergom 1979 (vol.I pp. 351–359) North-Holland: Amsterdam 1981Google Scholar
  7. [Fr2] Fritz, J.: Stationary measures of stochastic gradient systems, Infinite lattice models. Z. Wahrscheinlichkeitstheor. Verw. Geb.59, 479–490 (1982)Google Scholar
  8. [Fr3] Fritz, J.: On the stationary measures of anharmonic systems in the presence of a small thermal noise. J. Statist. Phys.44, 25–47 (1986)Google Scholar
  9. [Fr4] Fritz, J.: On the diffusive nature of entropy flow in infinite systems: Remarks to a paper by Guo-Papanicolau-Varadhan. Commun. Math. Phys.133, 331–352 (1990)Google Scholar
  10. [Fu1] Funaki, T.: Hydrodynamic limit for Ginzburg-Landau type continuum model. In: Grigelionis, et al. (eds.) Probability theory and mathematical statistics. (Proc. Fifth. Vilnius Conf. 1989, pp. 382–390)Google Scholar
  11. [Fu2] Funaki, T.: The reversible measures of multi-dimensional Ginzburg-Landau continuum model. Osaka J. Math.28, 463–494 (1991)Google Scholar
  12. [GV] Gallavotti, G., Verboven, E.: On the classical KMS condition. Nuovo Cimento, 274–286 (1976)Google Scholar
  13. [G] Georgii, H.-O.: Canonical Gibbs Measures (Lect. Notes Math., vol. 760) Berlin Heidelberg New York: Springer 1979Google Scholar
  14. [He] Hermann, R.: On the accessibility problem in control theory, II. International Symposium on Nonlinear Differential Equations and Nonlinear Mechanics. New York: Academic Press, pp. 325–332 (1963)Google Scholar
  15. [Ho] Holley, R.: Free energy in a Markovian model of a lattice spin system. Commun. Math. Phys.23, 87–99 (1971)Google Scholar
  16. [IK] Ichihara, K., Kunita, H.: A classification of the second order degenerate elliptic operators and its probabilistic characterization. Z. Wahrscheinlichkeitstheor. Verw. Geb.30, 235–254 (1974). Supplements and corrections. Z. Wahrscheinlichkeitstheor. Verw. Geb.39, 81–84 (1977)Google Scholar
  17. [K] Kumano-go, H.: Pseudo-differential operators. MIT Press 1981Google Scholar
  18. [LLL] Lanford, O.E., Lebowitz, J.L., Lieb, E.: Time evolution of infinite anharmonic systems. J. Statist. Phys.16, 453–461 (1977)Google Scholar
  19. [LP] Lebowitz, J.L., Presutti, E.: Statistical mechanics of systems of unbounded spins. Commun. Math. Phys.50, 195–218 (1976)Google Scholar
  20. [LS] Lebowitz, J.L., Spohn, H.: Stationary non-equilibrium states of infinite harmonic systems. Commun. Math. Phys.54, 97–121 (1977)Google Scholar
  21. [L] Liggett, T.M.: Interacting particle systems. Berlin Heidelberg New York: Springer 1985Google Scholar
  22. [OV] Olla, S., Varadhan, S.R.S.: Scaling limit for interacting Ornstein-Uhlenbeck processes. Commun. Math. Phys.135, 355–378 (1991)Google Scholar
  23. [OVY] Olla, S., Yau, H.T., Varadhan, S.R.S.: Hydrodynamic limit for Hamiltonian System with weak noise. Commun. Math. Phys.155, 523–560 (1993)Google Scholar
  24. [R] Ruelle, D.: Statistical Mechanics. Rigorous Results. Benjamin 1968Google Scholar
  25. [Si] Sinai, Ya.G.: Dynamics of local equilibrium Gibbs distributions and Euler equations. The one-dimensional case. Sel. Math. Sov.7, 279–289 (1987)Google Scholar
  26. [Sp] Spohn, H.: Large scale dynamics of interacting particles. Berlin Heidelberg New York: Springer 1991Google Scholar
  27. [SS] Shiga, T., Shimizu, A.: Infinite dimensional stochastic differential equations and their applications. J. Math. Kyoto Univ.20, 395–496 (1980)Google Scholar
  28. [St] Sternberg, S.: Lectures on differential geometry. Prentice-Hall 1964Google Scholar
  29. [SJ] Sussmann, H.J., Jurdjevic, V.: Controllability of nonlinear systems. J. Differ. Equations12, 95–116 (1972)Google Scholar
  30. [Y] Yau, H.T.: Relative entropy and hydrodynamics of Ginzburg-Landau models. Lett. Math. Phys.22, 63–80 (1991)Google Scholar

Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • J. Fritz
    • 1
  • T. Funaki
    • 2
  • J. L. Lebowitz
    • 3
  1. 1.Mathematical InstituteHungarian Academy of SciencesBudapestHungary
  2. 2.Department of Mathematics, Faculty of ScienceNagoya UniversityNagovaJapan
  3. 3.Department of Mathematics and PhysicsRutgers UniversityNew BrunswickUSA

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