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Smooth Dynamics and New Theoretical Ideas in Nonequilibrium Statistical Mechanics

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Abstract

This paper reviews various applications of the theory of smooth dynamical systems to conceptual problems of nonequilibrium statistical mecanics. We adopt a new point of view which has emerged progressively in recent years, and which takes seriously into account the chaotic character of the microscopic time evolution. The emphasis is on nonequilibrium steady states rather than the traditional approach to equilibrium point of view of Boltzmann. The nonequilibrium steady states, in presence of a Gaussian thermostat, are described by SRB measures. In terms of these one can prove the Gallavotti–Cohen fluctuation theorem. One can also prove a general linear response formula and study its consequences, which are not restricted to near-equilibrium situations. At equilibrium one recovers in particular the Onsager reciprocity relations. Under suitable conditions the nonequilibrium steady states satisfy the pairing theorem of Dettmann and Morriss. The results just mentioned hold so far only for classical systems; they do not involve large size, i.e., they hold without a thermodynamic limit.

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REFERENCES

  1. L. Andrey, The rate of entropy change in non-Hamiltonian systems, Phys. Lett. A 11:45–46 (1985).

    Google Scholar 

  2. L. Arnold, Random dynamical systems (Springer, Berlin, to appear).

  3. J. Bahnmüller and P.-D. Liu, Characterization of measures satisfying Pesin's entropy formula for random dynamical systems, J. Dynam. Diff. Eq. 10:425–448 (1998).

    Google Scholar 

  4. V. Baladi, Periodic orbits aud dynamical spectra (survey), Ergod. Th. and Dynam. Syst. 18:255–292 (1998).

    Google Scholar 

  5. L. Barreira, Ya. Pesin, and J. Schmeling, Dimension of hyperbolic measures—a proof of the Eckmann-Ruelle conjecture, Ann. of Math., to appear.

  6. P. Billingsley, Ergodic theory and information (John Wiley, New York, 1965).

    Google Scholar 

  7. F. Bonetto and G. Gallavotti, Reversibility, coarse graining and the chaoticity principle, Commun. Math. Phys. 189:263–276 (1997).

    Google Scholar 

  8. F. Bonetto, G. Gallavotti, and P. L. Garrido, Chaotic principle: An experimental test, Physica D 105:226–252 (1997).

    Google Scholar 

  9. R. Bowen, Markov partitions for Axiom A diffeomorphisms, Amer. J. Math. 92:725–747 (1970).

    Google Scholar 

  10. R. Bowen, Periodic points and measures for Axiom A diffeomorphisms, Trans. Amer. Math. Soc. 154:377–397 (1971).

    Google Scholar 

  11. R. Bowen, Periodic orbits for hyperbolic flows, Amer. J. Math. 94:1–30 (1972).

    Google Scholar 

  12. R. Bowen, Symbolic dynamics for hyperbolic flows, Amer. J. Math. 95:429–460 (1973).

    Google Scholar 

  13. R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Math., Vol. 470 (Springer, Berlin, 1975).

    Google Scholar 

  14. R. Bowen and D. Ruelle, The ergodic theory of Axiom A flows, Invent. Math. 29:181–202 (1975).

    Google Scholar 

  15. N. Chernov, Decay of correlations and dispersing billiards, J. Statist. Phys. 94:513–556 (1999).

    Google Scholar 

  16. N. I. Chernov, G. L. Eyink, J. L. Lebowitz, and Ya. G. Sinai, Derivation of Ohms law in a deterministic mechanical model, Phys. Rev. Lett. 70:2209–2212 (1993).

    Google Scholar 

  17. N. I. Chernov, G. L. Eyink, J. L. Lebowitz, and Ya. G. Sinai, Steady-state electrical conduction in the periodic Lorentz gas, Commun. Math. Phys. 154:569–601 (1993).

    Google Scholar 

  18. Ph. Choquard, Lagrangian formulation of Nosé-Hoover and of isokinetic dynamics, ESI report (1996).

  19. E. G. D. Cohen and L. Rondoni, Note on phase space contraction and entropy production in thermostatted Hamiltonian systems, Chaos 8:357–365 (1998).

    Google Scholar 

  20. S. R. de Groot and P. Mazur, Nonequilibrium thermodynamics (Dover, New York, 1984).

    Google Scholar 

  21. C. P. Dellago, H. A. Posch, and W. G. Hoover, Lyapunov instability in a system of hard disks in equilibrium and nonequilibrium steady states, Phys. Rev. E 53:1483–1501 (1996).

    Google Scholar 

  22. C. P. Dettmann and G. P. Morriss, Proof of Lyapunov exponent pairing for systems at constant kinetic energy, Phys. Rev. E 53:R5541–5544 (1996).

    Google Scholar 

  23. C. P. Dettmann and G. P. Morriss, Hamiltonian formulation of the Gaussian isokinetic thermostat, Phys. Rev. E 54:2495–2500 (1996).

    Google Scholar 

  24. C. P. Dettmann and G. P. Morriss, Hamiltonian reformulation and pairing of Lyapunov exponents for Nosé-Hoover dynamics, Phys. Rev. E 55:3693–3696 (1997).

    Google Scholar 

  25. D. Dolgopyat, On decay of correlations in Anosov flows, Ann. of Math. 147:357–390 (1998).

    Google Scholar 

  26. D. Dolgopyat, Prevalence of rapid mixing for hyperbolic flows, Ergod. Th. and Dynam. Syst. 18:1097–1114 (1998).

    Google Scholar 

  27. D. Dolgopyat and M. Pollicott, Addendum to “Periodic orbits and dynamical spectra,” Ergod. Th. and Dynam. Syst. 18:293–301 (1998).

    Google Scholar 

  28. J. R. Dorfman, An introduction to chaos in non-equilibrium statistical mechanics (Springer, Berlin, to appear).

  29. 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, Commun. Math. Phys., to appear.

  30. J.-P. Eckmann, C.-A. Pillet, and L. Rey-Bellet, Entropy production in non-linear, thermally driven Hamiltonian systems, Preprint.

  31. D. J. Evans, E. G. D. Cohen, and G. P. Morriss, Probability of second law violations in shearing steady states, Phys. Rev. Lett. 71:2401–2404 (1993).

    Google Scholar 

  32. D. J. Evans and G. P. Morriss, Statistical mechanics of nonequilibrium fluids (Academic Press, New York, 1990).

    Google Scholar 

  33. D. J. Evans and S. Sarman, Equivalence of thermostatted nonlinear responses, Phys. Rev. E 48:65–70 (1993).

    Google Scholar 

  34. J. Franks and R. F. Williams, Anomalous Anosov flows, pp. 158–174 in Lecture Notes in Mathematics, Vol. 819 (Springer, Berlin, 1980).

    Google Scholar 

  35. P. Frederickson, J. L. Kaplan, E. D. Yorke, and J. A. Yorke, The Lyapunov dimension of strange attractors, J. Diff. Eq. 49:185–207 (1983).

    Google Scholar 

  36. G. Gallavotti, Reversible Anosov diffeomorphisms and large deviations, Math. Phys. Electronic J. 1:1–12 (1995).

    Google Scholar 

  37. G. Gallavotti, Extension of Onsager's reciprocity to large fields and the chaotic hypothesis, Phys. Rev. Lett. 77:4334–4337 (1996).

    Google Scholar 

  38. G. Gallavotti, Chaotic hypothesis: Onsager reciprocity and fluctuation-dissipation theorem, J. Statist. Phys. 84:899–926 (1996).

    Google Scholar 

  39. G. Gallavotti, Equivalence of dynamical ensembles and Navier-Stokes equations, Phys. Lett. 223:91–95 (1996).

    Google Scholar 

  40. G. Gallavotti, Dynamical ensembles equivalence in fluid mechanics, Physica D 105:163–184 (1997).

    Google Scholar 

  41. G. Gallavotti, Chaotic dynamics, fluctuations, non-equilibrium ensembles, Chaos 8:384–392 (1998).

    Google Scholar 

  42. G. Gallavotti, New methods in nonequilibrium gases and fluids, in Proceedings of the conference Let's face chaos through nonlinear dynamics, M. Robnik, ed. (University of Maribor, 1996), Open Systems and Information Dynamics, to appear.

  43. G. Gallavotti, Fluctuation patterns and conditional reversibility in nonequilibrium systems, Ann. Inst. H. Poincaré, to appear (chao-dyn@xyz.lanl.gov #9703007).

  44. G. Gallavotti and E. G. D. Cohen, Dynamical ensembles in nonequilibrium statistical mechanics, Phys. Rev. Lett. 74:2694–2697 (1995).

    Google Scholar 

  45. G. Gallavotti and E. G. D. Cohen, Dynamical ensembles in stationary states, J. Statist. Phys. 80:931–970 (1995).

    Google Scholar 

  46. G. Gallavotti and D. Ruelle, SRB states and nonequilibrium statistical mechanics close to equilibrium, Commun. Math. Phys. 190:279–281 (1997).

    Google Scholar 

  47. P. Gaspard and J. R. Dorfman, Phys. Rev. E 52:3525–3552 (1995).

    Google Scholar 

  48. P. Gaspard and G. Nicolis, Transport properties, Lyapunov exponents, and entropy per unit time, Phys. Rev. Lett. 65:1693–1696 (1990).

    Google Scholar 

  49. G. Gentile, Large deviation rule for Anosov flows, Forum Math. 10:89–118 (1998).

    Google Scholar 

  50. M. S. Green, Brownian motion in a gas of noninteracting molecules, J. Chem. Phys. 19:1036–1046 (1951).

    Google Scholar 

  51. W. G. Hoover, Molecular dynamics, Lecture Notes in Physics, Vol. 258 (Springer, Heidelberg, 1986).

    Google Scholar 

  52. W. G. Hoover, B. Moran, C. Hoover, and W. J. Evans, Irreversibility in the Galton board via conservative classical and quantum Hamiltonian and Gaussian dynamics, Phys. Lett. 133:114–120 (1988).

    Google Scholar 

  53. V. Jakšićand C.-A. Pillet, Ergodic properties of classical dissipative systems I, Acta Mathematica, to appear.

  54. J. L. Kaplan and J. A. Yorke, Preturbulence: A regime observed in a fluid flow of Lorenz, Commun. Math. Phys. 67:93–108 (1979).

    Google Scholar 

  55. Yu. Kifer, Ergodic theory of random transformations (Birkhäuser, Boston, 1986).

    Google Scholar 

  56. Yu. Kifer, Random perturbations of dynamical systems (Birkhäuser, Boston, 1988).

    Google Scholar 

  57. R. Kubo, Statistical-mechanical theory of irreversible processes. I, J. Phys. Soc. (Japan) 12:570–586 (1957).

    Google Scholar 

  58. J. Kurchan, Fluctuation theorem for stochastic dynamics, J. Phys. A: Math. Gen. 31:3719–3729 (1998).

    Google Scholar 

  59. O. E. Lanford, Entropy and equilibrium states in classical statistical mechanics, pp. 1–113 in Lecture Notes in Physics, Vol. 20 (Springer, Berlin, 1973).

    Google Scholar 

  60. O. E. Lanford, Time evolution of large classical systems, pp. 1–111 in Lecture Notes in Physics, Vol. 38 (Spinger, Berlin, 1975).

    Google Scholar 

  61. A. Latz, H. van Beijeren, and J. R. Dorfman, Lyapunov spectrum and the conjugate pairing rule for a thermostatted random gas: kinetic theory, Phys. Rev. Lett. 78:207–210 (1997).

    Google Scholar 

  62. J. L. Lebowitz, Boltzwann's entropy and time's arrow, Physics Today 46(9):32–38 (1993).

    Google Scholar 

  63. J. L. Lebowitz and H. Spohn, Transport properties of the Lorentz gas: Fourier's law, J. Statist. Phys. 19:633–654 (1978).

    Google Scholar 

  64. J. L. Lebowitz and H. Spohn, A Gallavotti-Cohen type fluctuation theorem for stochastic dynamics, Preprint.

  65. F. Ledrappier, Propriétés ergodiques des mesures de Sinai, Publ. Math. IHES 59:163–188 (1984).

    Google Scholar 

  66. F. Ledrappier and J.-M. Strelcyn, A proof of the estimation from below in Pesin's entropy formula, Ergod. Th. and Dynam. Syst. 2:203–219 (1982).

    Google Scholar 

  67. F. Ledrappier and L. S. Young, The metric entropy of diffeomorphisms: I. Characterization of measures satisfying Pesin's formula, II. Relations between entropy, exponents and dimension, Ann. of Math. 122:509-539, 540-574 (1985).

    Google Scholar 

  68. L. Onsager, Reciprocal relations in irreversible processes. II, Phys. Rev. 38:2265–2279 (1931).

    Google Scholar 

  69. V. I. Oseledec, A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems, Tr. Mosk. Mat. Obšč 19:179–210 (1968). English translation, Trans. Moscow Math. Soc. 19:197-221 (1968).

    Google Scholar 

  70. W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque 187–188, Soc. Math. de France, Paris, 1990.

  71. Ya. B. Pesin, Invariant manifold families which correspond to non-vanishing characteristic exponents, Izv. Akad. Nauk SSSR. Ser. Mat. 40(6):1332–1379 (1976). English translation, Math. USSR Izv. 10(6):1261-1305 (1976).

    Google Scholar 

  72. Ya. B. Pesin, Lyapunov characteristic exponents and smooth ergodic theory, Uspehi Mat. Nauk 32(4):55–112 (1977). English translation, Russian Math. Surveys. 32(4):55-114 (1977).

    Google Scholar 

  73. H. A. Posch and W. G. Hoover, Lyapunov instability in dense Lennard-Jones fluids, Phys. Rev. A 38:473–482 (1988).

    Google Scholar 

  74. I. Prigogine, Introduction to thermodynamics of irreversible processes (John Wiley, New York, 1962).

    Google Scholar 

  75. C. C. Pugh and M. Shub, Ergodic attractors, Trans. Amer. Math. Soc. 312:1–54 (1989).

    Google Scholar 

  76. M. Ratner, Markov partitions for Anosov flows on 3-dimensional manifolds, Mat. Zam. 6:693–704 (1969).

    Google Scholar 

  77. Z. Rieder, J. L. Lebowitz, and E. Lieb, Properties of a harmonic crystal in a stationary nonequilibrium state, J. Math. Phys. 8:1073–1078 (1967).

    Google Scholar 

  78. D. Ruelle, Correlation functionals, J. Math. Phys. 6:201–220 (1965).

    Google Scholar 

  79. D. Ruelle, A measure associated with Axiom A attractors, Amer. J. Math. 98:619–654 (1976).

    Google Scholar 

  80. D. Ruelle, An inequality for the entropy of differentiable maps, Bol. Soc. Bras. Mat. 9:83–87 (1978).

    Google Scholar 

  81. D. Ruelle, Thermodynamic formalism (Addison-Wesley, Reading, MA, 1978).

    Google Scholar 

  82. D. Ruelle, Ergodic theory of differentiable dynamical systems, Publ. Math. IHES 50:27–58 (1979).

    Google Scholar 

  83. D. Ruelle, One-dimensional Gibbs states and Axiom A diffeomorphisms, J. Diff. Geom. 25:117–137 (1987).

    Google Scholar 

  84. D. Ruelle, Resonances for Axiom A flows, J. Diff. Geom. 25:99–116 (1987).

    Google Scholar 

  85. D. Ruelle, Positivity of entropy production in nonequilibrium statistical mechanics, J. Satist. Phys. 85:1–23 (1996).

    Google Scholar 

  86. D. Ruelle, Differentiation of SRB states, Commun. Math. Phys. 187:227–241 (1997).

    Google Scholar 

  87. D. Ruelle, Nonequillibrium statistical mechanics near equilibrium: computing higher order terms, Nonlinearity 11:5–18 (1998).

    Google Scholar 

  88. D. Ruelle, General linear response formula in statistical mechanics, and the fluctuation-dissipation theorem far from equilibrium, Phys. Lett. A 245:220–224 (1998).

    Google Scholar 

  89. D. J. Searles, D. J. Evans, and D. J. Isbister, The number dependence of the maximum Lyapunov exponent, Physica A 240:96–104 (1977).

    Google Scholar 

  90. Ya. G. Sinai, Markov partitions and C-diffeomorphisms, Funkts. Analiz i Ego Pril. 2(1):64–89 (1968). English translation, Functional Anal. Appl. 2:61-82 (1968).

    Google Scholar 

  91. Ya. G. Sinai, Constuction of Markov partitions, Funkts. Analiz i Ego Pril. 2(3):70–80 (1968). English translation, Functional Anal. Appl. 2:245-253 (1968).

    Google Scholar 

  92. Ya. G. Sinai, Gibbsian measures in ergodic theory, Uspehi Mat. Nauk 27(4):21–64 (1972). English translation, Russian Math. Surveys 27(4):21-69 (1972).

    Google Scholar 

  93. S. Smale, Differentiable dynamical systems, Bull. AMS 73:747–817 (1967).

    Google Scholar 

  94. H. Spohn and S. L. Lebowitz, Stationary non-equilibrium states of infinite harmonic systems, Commun. Math. Phys. 54:97–120 (1977).

    Google Scholar 

  95. M. Viana, Multidimensional nonhyperbolic attractors, Publ. Math. IHES 85:63–96 (1997).

    Google Scholar 

  96. M. P. Wojtkowski and C. Liverani, Conformally symplectic dynamics and symmetry of the Lyapunov spectrum, Commun. Math. Phys. 194:47–60 (1998).

    Google Scholar 

  97. L.-S. Young, Statistical properties of dynamical systems with some hyperbolicity, Ann. of Math. 147:585–650 (1998).

    Google Scholar 

  98. L.-S. Young, Recurrence times and rates of mixing, Preprint.

  99. L.-S. Young, Ergodic theory of chaotic dynamical systems, Lecture at the International Congress of Mathematical Physics, 1997.

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  1. IHES, Bures sur Yvette, France, and Department of Mathematics, Rutgers University, Brunswick, New Jersey

    David Ruelle

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Ruelle, D. Smooth Dynamics and New Theoretical Ideas in Nonequilibrium Statistical Mechanics. Journal of Statistical Physics 95, 393–468 (1999). https://doi.org/10.1023/A:1004593915069

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