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Journal of Statistical Physics

, Volume 95, Issue 1–2, pp 333–365 | Cite as

A Gallavotti–Cohen-Type Symmetry in the Large Deviation Functional for Stochastic Dynamics

  • Joel L. Lebowitz
  • Herbert Spohn
Article

Abstract

We extend the work of Kurchan on the Gallavotti–Cohen fluctuation theorem, which yields a symmetry property of the large deviation function, to general Markov processes. These include jump processes describing the evolution of stochastic lattice gases driven in the bulk or through particle reservoirs, general diffusive processes in physical and/or velocity space, as well as Hamiltonian systems with stochastic boundary conditions. For dynamics satisfying local detailed balance we establish a link between the average of the action functional in the fluctuation theorem and the macroscopic entropy production. This gives, in the linear regime, an alternative derivation of the Green–Kubo formula and the Onsager reciprocity relations. In the nonlinear regime consequences of the new symmetry are harder to come by and the large deviation functional difficult to compute. For the asymmetric simple exclusion process the latter is determined explicitly using the Bethe ansatz in the limit of large N.

fluctuation theorem current fluctuations asymmetric exclusion process 

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REFERENCES

  1. 1.
    P. G. Bergmann and J. L. Lebowitz, New approach to nonequlibrium processes, Phys. Rev. 99:578 (1955).Google Scholar
  2. 2.
    J. L. Lebowitz and P. G. Bergmann, Irreversible Gibbsian ensembles, Ann. Phys. 1:1 (1957).Google Scholar
  3. 3.
    Z. Rieder, J. L. Lebowitz, and E. Lieb, Properties of a harmonic crystal in a stationary nonequilibrium state, J. Math. Phys. 8:1073 (1967).Google Scholar
  4. 4.
    A. J. O'Conner and J. L. Lebowitz, Heat conduction and sound transmission in isotopically disordered harmonic crystals, Journ. Math. Phys. 15:629 (1974).Google Scholar
  5. 5.
    S. Goldstein, J. L. Lebowitz, and E. Presutti, Mechanical system with stochastic boundaries, in Random Fields, Vol. I, J. Fritz, J. L. Lebowitz, and D. Szász, eds. (North-Holland, Amsterdam, 1979).Google Scholar
  6. 6.
    J. L. Lebowitz, Exact results in nonequilibrium statistical mechanics: Where do we stand?, Prog. Theor. Physics, Supplement 64:35 (1978).Google Scholar
  7. 7.
    S. Goldstein, C. Kipnis, and N. Ianiro, Stationary states for a mechanical system with stochastic boundary conditions, J. Stat. Phys. 41:915 (1985).Google Scholar
  8. 8.
    H. Spohn, and J. L. Lebowitz, Stationary nonequilibrium states of infinite harmonic systems, Comm. Math. Phys. 54:97 (1977).Google Scholar
  9. 9.
    S. Goldstein, J. L. Lebowitz, and K. Ravishankar, Approach to equilibrium in models of a system in contact with a heat bath, J. Stat. Phys. 43:303 (1986).Google Scholar
  10. 10.
    J.-P. Eckmann, C.-A. Pillet, and L. Rey-Bellet, Non-equilibrium statistical mechanics of anharmonic chains coupled to two heat baths at different temperatures, preprint 1998, Texas Archive for Mathematical Physics.Google Scholar
  11. 11.
    S. Katz, J. L. Lebowitz, and H. Spohn, Stationary nonequilibrium states for stochastic lattice gas models of ionic superconductors, J. Stat. Phys. 34:497 (1984).Google Scholar
  12. 12.
    H. Spohn, Long range correlations for stochastic lattice gases in a nonequilibrium steady state, J. Phys. A 16:4275 (1983).Google Scholar
  13. 13.
    G. Eyink, J. L. Lebowitz, and H. Spohn, Microscopic origin of hydrodynamic behavior: Entropy production and the steady state, in Chaos, Soviet-American Perspectives in Non-linear Science, Hg. D. K Campbell, (American Institute of Physics, 1990), p. 367.Google Scholar
  14. 14.
    G. Eyink, J. L. Lebowitz, and H. Spohn, Hydrodynamics, fluctuations, and large deviations outside local equilibrium, J. Stat. Phys. 83:385 (1996).Google Scholar
  15. 15.
    W.G. Hoover, Molecular Dynamics, Lecture Notes in Physics, Vol. 258 (Springer, Heidelberg, 1986).Google Scholar
  16. 16.
    D. J. Evans and G. P. Morriss, Statistical Mechanics of Nonequilibrium Fluids (Academic Press, London, 1990).Google Scholar
  17. 17.
    N. J. Chernov, G. L. Eyink, J. L. Lebowitz, and Ya. G. Sinai, Derivation of Ohm's law in a determinisitic mechanical model, Phys. Rev. Lett. 70:2209 (1993).Google Scholar
  18. 18.
    N. J. Chernov, G. L. Eyink, J. L. Lebowitz, and Ya. G. Sinai, Steady-state electrical conduction in the periodic Lorentz gas, Comm. Math. Phys. 154:569 (1993).Google Scholar
  19. 19.
    D. Ruelle, Positivity of entropy production in nonequilibrium statistical mechanics, J. Stat. Phys. 85:1 (1996).Google Scholar
  20. 20.
    G. Gallavotti, Chaotic dynamics, fluctuations, non-equilibrium ensembles, Chaos 8:384 (1998).Google Scholar
  21. 21.
    D. Ruelle, New theoretical ideas in nonequilibrum statistical mechanics, Lecture Notes (Rutgers University, fall 1997).Google Scholar
  22. 22.
    G. Gallavotti, and E. G. D. Cohen, Dynamical ensembles in stationary states, J. Stat. Phys. 80:931 (1995).Google Scholar
  23. 23.
    D. J. Evans, E. G. D. Cohen, and G. P. Morriss, Probability of second law violations in steady flows, Phys. Rev. Lett. 71:2401 (1993).Google Scholar
  24. 24.
    G. Gallavotti, Extension of Onsager's reciprocity to large fields and the chaotic hypothesis, Phys. Rev. Lett. 77:4334 (1996).Google Scholar
  25. 25.
    G. Gallavotti, New methods in nonequilibrium gases and fluids, Proceedings of the conference “Let's face chaos through nonlinear dynamics”, University of Maribor, 24 june-5 july 1996, M. Robnik, ed. Open Systems and Information Dynamics, Vol. 5, 1998, to be published. Archived as: chao-dyn 9610018.Google Scholar
  26. 26.
    D. J. Evans and D. J. Searles, Equilibrium microstates which generate second law violating steady states, Phys. Rev. E 50:1645 (1994).Google Scholar
  27. 27.
    J. Kurchan, Fluctuation theorem for stochastic dynamics, J. Phys. A.: Math. Gen. 31:3719 (1998).Google Scholar
  28. 28.
    F. Bonetto, G. Gallavotti, and P. Garrido, Chaotic principle: an experimental test, Physica D 105:226 (1997).Google Scholar
  29. 29.
    C. Maes, The fluctuation theorem as a Gibbs property, preprint, 1998.Google Scholar
  30. 30.
    S. R. S. Varadhan, Large Deviations and Applications (SIAM, Philadelphia, 1984).Google Scholar
  31. 31.
    J.-D. Deuschel and D. W. Strook, Large Deviations (Academic Press, San Diego, 1989).Google Scholar
  32. 32.
    H. Spohn, Large Scale Dynamics of Interacting Particles (Springer, Heidelberg, 1991).Google Scholar
  33. 33.
    D. Stroock and S. R. S. Varadhan, Multidimensional Diffusion Processes (Springer, Berlin, 1979).Google Scholar
  34. 34.
    M. Büttiker, Transport as a consequence of state-dependent diffusion, Z. Physik 68:161 (1987).Google Scholar
  35. 35.
    Ya. M. Blanter and M. Büttiker, Rectification of fluctuations in an underdamped ratchet, preprint 1998.Google Scholar
  36. 36.
    J.-P. Eckmann, C.-A. Pillet, and L. Rey-Bellet, Entropy production in non-linear, thermally driven Hamiltonian systems, J. Stat. Phys., to appear; Nonequilibrium statistical mechanics of anharmonic chains coupled to two heat baths at different temperatures, Commun. Math. Phys., to appear.Google Scholar
  37. 37.
    B. L. Holian, W. G. Hoover, and H. A. Posch, Resolution of Loschmidt's paradox: The origin of irreversible behavior in reversible atomistic dynamics, Phys. Rev. Lett. 59:10 (1987).Google Scholar
  38. 38.
    H. A. Posch and W. G. Hoover, Non equilibrium molecular dynamics of a classical fluid, in Molecular Liquids: New Perspectives in Physics and Chemistry, J. Teixeira-Dias, ed. (Kluwer Academic Publishers, 1992), p. 527.Google Scholar
  39. 39.
    G. Gallavotti, Chaotic hypothesis: Onsager reciprocity and fluctuation dissipation theorem, J. Stat. Phys. 84:899 (1996).Google Scholar
  40. 40.
    S. Lipri, R. Livi, and A. Politi, Energy transport in anharmonic lattice close and far from equilibrium, preprint, archived in xxx.lanl.gov.cond-mat #9709195.Google Scholar
  41. 41.
    N. I. Chernov and J. L. Lebowitz, Stationary nonequilibrium states in boundary-driven Hamiltonian systems: Shear flow, J. Stat. Phys. 86:953 (1997).Google Scholar
  42. 42.
    F. Bonetto, N. I. Chernov, and J. L. Lebowitz, (Global and local) fluctuations of phase space contraction in deterministic stationary non-equilibrium, Chaos 8:823–833 (1998).Google Scholar
  43. 43.
    F. Bonetto and J. L. Lebowitz (work in progress).Google Scholar
  44. 44.
    G. Gallavotti and D. Ruelle, SNOB states and nonequilibrium statistical mechanics close to equilibrium, Comm. Math. Phys. 190:279 (1997).Google Scholar
  45. 45.
    D. Ruelle, Differentiation of SRB states, Comm. Math. Phys. 187:227 (1997).Google Scholar
  46. 46.
    B. Suthertand, C. N. Yang, and C. P. Yang, Exact solution of a model of two-dimensional ferroelectric in an arbitrary external electric field, Phys. Rev. Lett. 19:588 (1967).Google Scholar
  47. 47.
    D. Kim, Bethe ansatz solution for crossover scaling functions of the asymmetric XXZ chain and the KPZ-type growth model, Phys. Rev. E 52:3512 (1995).Google Scholar
  48. 48.
    B. Derrida and J. L. Lebowitz, Exact large deviation function in the asymmetric exclusion process, Phys. Rev. Lett. 80:209 (1998).Google Scholar
  49. 49.
    B. Derrida and C. Appert, Universal large deviation function of the Kardar-Parisi-Zhang equation in one dimension, preprint, 1998.Google Scholar
  50. 50.
    H. van Beijeren, R. Kutner, and H. Spohn, Excess noise for driven diffusive systems, Phys. Rev. Lett. 54:2026 (1985).Google Scholar
  51. 51.
    H. van Beijeren, Transport properties of stochastic Lorentz models, Rev. Mod. Phys. 54:195 (1982).Google Scholar
  52. 52.
    C. N. Yang and C. P. Yang, Ground state energy of a Heisenberg-Ising lattice, Phys. Rev. 147:303 (1966).Google Scholar
  53. 53.
    G. Gallavotti, Fluctuation patterns and conditional reversibility in nonequilibrium systems, Ann. Institut H. Poincaré, in print, and chao-dyn@xyz.lanl.gov #9703007.Google Scholar
  54. 54.
    G. Gallavotti, private communication.Google Scholar

Copyright information

© Kluwer Academic/Plenum Publishers 1999

Authors and Affiliations

  • Joel L. Lebowitz
    • 1
  • Herbert Spohn
    • 2
  1. 1.Department of Mathematics and PhysicsRutgers University, Hill CenterNew Brunswick
  2. 2.Zentrum Mathematik and Physik Department, TU MünchenMünchenGermany

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