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
L. Andrey, The rate of entropy change in non-Hamiltonian systems, Phys. Lett. A 11:45–46 (1985).
L. Arnold, Random dynamical systems (Springer, Berlin, to appear).
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).
V. Baladi, Periodic orbits aud dynamical spectra (survey), Ergod. Th. and Dynam. Syst. 18:255–292 (1998).
L. Barreira, Ya. Pesin, and J. Schmeling, Dimension of hyperbolic measures—a proof of the Eckmann-Ruelle conjecture, Ann. of Math., to appear.
P. Billingsley, Ergodic theory and information (John Wiley, New York, 1965).
F. Bonetto and G. Gallavotti, Reversibility, coarse graining and the chaoticity principle, Commun. Math. Phys. 189:263–276 (1997).
F. Bonetto, G. Gallavotti, and P. L. Garrido, Chaotic principle: An experimental test, Physica D 105:226–252 (1997).
R. Bowen, Markov partitions for Axiom A diffeomorphisms, Amer. J. Math. 92:725–747 (1970).
R. Bowen, Periodic points and measures for Axiom A diffeomorphisms, Trans. Amer. Math. Soc. 154:377–397 (1971).
R. Bowen, Periodic orbits for hyperbolic flows, Amer. J. Math. 94:1–30 (1972).
R. Bowen, Symbolic dynamics for hyperbolic flows, Amer. J. Math. 95:429–460 (1973).
R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Math., Vol. 470 (Springer, Berlin, 1975).
R. Bowen and D. Ruelle, The ergodic theory of Axiom A flows, Invent. Math. 29:181–202 (1975).
N. Chernov, Decay of correlations and dispersing billiards, J. Statist. Phys. 94:513–556 (1999).
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).
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).
Ph. Choquard, Lagrangian formulation of Nosé-Hoover and of isokinetic dynamics, ESI report (1996).
E. G. D. Cohen and L. Rondoni, Note on phase space contraction and entropy production in thermostatted Hamiltonian systems, Chaos 8:357–365 (1998).
S. R. de Groot and P. Mazur, Nonequilibrium thermodynamics (Dover, New York, 1984).
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).
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).
C. P. Dettmann and G. P. Morriss, Hamiltonian formulation of the Gaussian isokinetic thermostat, Phys. Rev. E 54:2495–2500 (1996).
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).
D. Dolgopyat, On decay of correlations in Anosov flows, Ann. of Math. 147:357–390 (1998).
D. Dolgopyat, Prevalence of rapid mixing for hyperbolic flows, Ergod. Th. and Dynam. Syst. 18:1097–1114 (1998).
D. Dolgopyat and M. Pollicott, Addendum to “Periodic orbits and dynamical spectra,” Ergod. Th. and Dynam. Syst. 18:293–301 (1998).
J. R. Dorfman, An introduction to chaos in non-equilibrium statistical mechanics (Springer, Berlin, to appear).
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.
J.-P. Eckmann, C.-A. Pillet, and L. Rey-Bellet, Entropy production in non-linear, thermally driven Hamiltonian systems, Preprint.
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).
D. J. Evans and G. P. Morriss, Statistical mechanics of nonequilibrium fluids (Academic Press, New York, 1990).
D. J. Evans and S. Sarman, Equivalence of thermostatted nonlinear responses, Phys. Rev. E 48:65–70 (1993).
J. Franks and R. F. Williams, Anomalous Anosov flows, pp. 158–174 in Lecture Notes in Mathematics, Vol. 819 (Springer, Berlin, 1980).
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).
G. Gallavotti, Reversible Anosov diffeomorphisms and large deviations, Math. Phys. Electronic J. 1:1–12 (1995).
G. Gallavotti, Extension of Onsager's reciprocity to large fields and the chaotic hypothesis, Phys. Rev. Lett. 77:4334–4337 (1996).
G. Gallavotti, Chaotic hypothesis: Onsager reciprocity and fluctuation-dissipation theorem, J. Statist. Phys. 84:899–926 (1996).
G. Gallavotti, Equivalence of dynamical ensembles and Navier-Stokes equations, Phys. Lett. 223:91–95 (1996).
G. Gallavotti, Dynamical ensembles equivalence in fluid mechanics, Physica D 105:163–184 (1997).
G. Gallavotti, Chaotic dynamics, fluctuations, non-equilibrium ensembles, Chaos 8:384–392 (1998).
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.
G. Gallavotti, Fluctuation patterns and conditional reversibility in nonequilibrium systems, Ann. Inst. H. Poincaré, to appear (chao-dyn@xyz.lanl.gov #9703007).
G. Gallavotti and E. G. D. Cohen, Dynamical ensembles in nonequilibrium statistical mechanics, Phys. Rev. Lett. 74:2694–2697 (1995).
G. Gallavotti and E. G. D. Cohen, Dynamical ensembles in stationary states, J. Statist. Phys. 80:931–970 (1995).
G. Gallavotti and D. Ruelle, SRB states and nonequilibrium statistical mechanics close to equilibrium, Commun. Math. Phys. 190:279–281 (1997).
P. Gaspard and J. R. Dorfman, Phys. Rev. E 52:3525–3552 (1995).
P. Gaspard and G. Nicolis, Transport properties, Lyapunov exponents, and entropy per unit time, Phys. Rev. Lett. 65:1693–1696 (1990).
G. Gentile, Large deviation rule for Anosov flows, Forum Math. 10:89–118 (1998).
M. S. Green, Brownian motion in a gas of noninteracting molecules, J. Chem. Phys. 19:1036–1046 (1951).
W. G. Hoover, Molecular dynamics, Lecture Notes in Physics, Vol. 258 (Springer, Heidelberg, 1986).
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).
V. Jakšićand C.-A. Pillet, Ergodic properties of classical dissipative systems I, Acta Mathematica, to appear.
J. L. Kaplan and J. A. Yorke, Preturbulence: A regime observed in a fluid flow of Lorenz, Commun. Math. Phys. 67:93–108 (1979).
Yu. Kifer, Ergodic theory of random transformations (Birkhäuser, Boston, 1986).
Yu. Kifer, Random perturbations of dynamical systems (Birkhäuser, Boston, 1988).
R. Kubo, Statistical-mechanical theory of irreversible processes. I, J. Phys. Soc. (Japan) 12:570–586 (1957).
J. Kurchan, Fluctuation theorem for stochastic dynamics, J. Phys. A: Math. Gen. 31:3719–3729 (1998).
O. E. Lanford, Entropy and equilibrium states in classical statistical mechanics, pp. 1–113 in Lecture Notes in Physics, Vol. 20 (Springer, Berlin, 1973).
O. E. Lanford, Time evolution of large classical systems, pp. 1–111 in Lecture Notes in Physics, Vol. 38 (Spinger, Berlin, 1975).
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).
J. L. Lebowitz, Boltzwann's entropy and time's arrow, Physics Today 46(9):32–38 (1993).
J. L. Lebowitz and H. Spohn, Transport properties of the Lorentz gas: Fourier's law, J. Statist. Phys. 19:633–654 (1978).
J. L. Lebowitz and H. Spohn, A Gallavotti-Cohen type fluctuation theorem for stochastic dynamics, Preprint.
F. Ledrappier, Propriétés ergodiques des mesures de Sinai, Publ. Math. IHES 59:163–188 (1984).
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).
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).
L. Onsager, Reciprocal relations in irreversible processes. II, Phys. Rev. 38:2265–2279 (1931).
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).
W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque 187–188, Soc. Math. de France, Paris, 1990.
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).
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).
H. A. Posch and W. G. Hoover, Lyapunov instability in dense Lennard-Jones fluids, Phys. Rev. A 38:473–482 (1988).
I. Prigogine, Introduction to thermodynamics of irreversible processes (John Wiley, New York, 1962).
C. C. Pugh and M. Shub, Ergodic attractors, Trans. Amer. Math. Soc. 312:1–54 (1989).
M. Ratner, Markov partitions for Anosov flows on 3-dimensional manifolds, Mat. Zam. 6:693–704 (1969).
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).
D. Ruelle, Correlation functionals, J. Math. Phys. 6:201–220 (1965).
D. Ruelle, A measure associated with Axiom A attractors, Amer. J. Math. 98:619–654 (1976).
D. Ruelle, An inequality for the entropy of differentiable maps, Bol. Soc. Bras. Mat. 9:83–87 (1978).
D. Ruelle, Thermodynamic formalism (Addison-Wesley, Reading, MA, 1978).
D. Ruelle, Ergodic theory of differentiable dynamical systems, Publ. Math. IHES 50:27–58 (1979).
D. Ruelle, One-dimensional Gibbs states and Axiom A diffeomorphisms, J. Diff. Geom. 25:117–137 (1987).
D. Ruelle, Resonances for Axiom A flows, J. Diff. Geom. 25:99–116 (1987).
D. Ruelle, Positivity of entropy production in nonequilibrium statistical mechanics, J. Satist. Phys. 85:1–23 (1996).
D. Ruelle, Differentiation of SRB states, Commun. Math. Phys. 187:227–241 (1997).
D. Ruelle, Nonequillibrium statistical mechanics near equilibrium: computing higher order terms, Nonlinearity 11:5–18 (1998).
D. Ruelle, General linear response formula in statistical mechanics, and the fluctuation-dissipation theorem far from equilibrium, Phys. Lett. A 245:220–224 (1998).
D. J. Searles, D. J. Evans, and D. J. Isbister, The number dependence of the maximum Lyapunov exponent, Physica A 240:96–104 (1977).
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).
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).
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).
S. Smale, Differentiable dynamical systems, Bull. AMS 73:747–817 (1967).
H. Spohn and S. L. Lebowitz, Stationary non-equilibrium states of infinite harmonic systems, Commun. Math. Phys. 54:97–120 (1977).
M. Viana, Multidimensional nonhyperbolic attractors, Publ. Math. IHES 85:63–96 (1997).
M. P. Wojtkowski and C. Liverani, Conformally symplectic dynamics and symmetry of the Lyapunov spectrum, Commun. Math. Phys. 194:47–60 (1998).
L.-S. Young, Statistical properties of dynamical systems with some hyperbolicity, Ann. of Math. 147:585–650 (1998).
L.-S. Young, Recurrence times and rates of mixing, Preprint.
L.-S. Young, Ergodic theory of chaotic dynamical systems, Lecture at the International Congress of Mathematical Physics, 1997.
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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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DOI: https://doi.org/10.1023/A:1004593915069