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.
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REFERENCES
P. G. Bergmann and J. L. Lebowitz, New approach to nonequlibrium processes, Phys. Rev. 99:578 (1955).
J. L. Lebowitz and P. G. Bergmann, Irreversible Gibbsian ensembles, Ann. Phys. 1:1 (1957).
Z. Rieder, J. L. Lebowitz, and E. Lieb, Properties of a harmonic crystal in a stationary nonequilibrium state, J. Math. Phys. 8:1073 (1967).
A. J. O'Conner and J. L. Lebowitz, Heat conduction and sound transmission in isotopically disordered harmonic crystals, Journ. Math. Phys. 15:629 (1974).
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).
J. L. Lebowitz, Exact results in nonequilibrium statistical mechanics: Where do we stand?, Prog. Theor. Physics, Supplement 64:35 (1978).
S. Goldstein, C. Kipnis, and N. Ianiro, Stationary states for a mechanical system with stochastic boundary conditions, J. Stat. Phys. 41:915 (1985).
H. Spohn, and J. L. Lebowitz, Stationary nonequilibrium states of infinite harmonic systems, Comm. Math. Phys. 54:97 (1977).
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).
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.
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).
H. Spohn, Long range correlations for stochastic lattice gases in a nonequilibrium steady state, J. Phys. A 16:4275 (1983).
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.
G. Eyink, J. L. Lebowitz, and H. Spohn, Hydrodynamics, fluctuations, and large deviations outside local equilibrium, J. Stat. Phys. 83:385 (1996).
W.G. Hoover, Molecular Dynamics, Lecture Notes in Physics, Vol. 258 (Springer, Heidelberg, 1986).
D. J. Evans and G. P. Morriss, Statistical Mechanics of Nonequilibrium Fluids (Academic Press, London, 1990).
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).
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).
D. Ruelle, Positivity of entropy production in nonequilibrium statistical mechanics, J. Stat. Phys. 85:1 (1996).
G. Gallavotti, Chaotic dynamics, fluctuations, non-equilibrium ensembles, Chaos 8:384 (1998).
D. Ruelle, New theoretical ideas in nonequilibrum statistical mechanics, Lecture Notes (Rutgers University, fall 1997).
G. Gallavotti, and E. G. D. Cohen, Dynamical ensembles in stationary states, J. Stat. Phys. 80:931 (1995).
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).
G. Gallavotti, Extension of Onsager's reciprocity to large fields and the chaotic hypothesis, Phys. Rev. Lett. 77:4334 (1996).
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.
D. J. Evans and D. J. Searles, Equilibrium microstates which generate second law violating steady states, Phys. Rev. E 50:1645 (1994).
J. Kurchan, Fluctuation theorem for stochastic dynamics, J. Phys. A.: Math. Gen. 31:3719 (1998).
F. Bonetto, G. Gallavotti, and P. Garrido, Chaotic principle: an experimental test, Physica D 105:226 (1997).
C. Maes, The fluctuation theorem as a Gibbs property, preprint, 1998.
S. R. S. Varadhan, Large Deviations and Applications (SIAM, Philadelphia, 1984).
J.-D. Deuschel and D. W. Strook, Large Deviations (Academic Press, San Diego, 1989).
H. Spohn, Large Scale Dynamics of Interacting Particles (Springer, Heidelberg, 1991).
D. Stroock and S. R. S. Varadhan, Multidimensional Diffusion Processes (Springer, Berlin, 1979).
M. Büttiker, Transport as a consequence of state-dependent diffusion, Z. Physik 68:161 (1987).
Ya. M. Blanter and M. Büttiker, Rectification of fluctuations in an underdamped ratchet, preprint 1998.
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.
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).
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.
G. Gallavotti, Chaotic hypothesis: Onsager reciprocity and fluctuation dissipation theorem, J. Stat. Phys. 84:899 (1996).
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.
N. I. Chernov and J. L. Lebowitz, Stationary nonequilibrium states in boundary-driven Hamiltonian systems: Shear flow, J. Stat. Phys. 86:953 (1997).
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).
F. Bonetto and J. L. Lebowitz (work in progress).
G. Gallavotti and D. Ruelle, SNOB states and nonequilibrium statistical mechanics close to equilibrium, Comm. Math. Phys. 190:279 (1997).
D. Ruelle, Differentiation of SRB states, Comm. Math. Phys. 187:227 (1997).
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).
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).
B. Derrida and J. L. Lebowitz, Exact large deviation function in the asymmetric exclusion process, Phys. Rev. Lett. 80:209 (1998).
B. Derrida and C. Appert, Universal large deviation function of the Kardar-Parisi-Zhang equation in one dimension, preprint, 1998.
H. van Beijeren, R. Kutner, and H. Spohn, Excess noise for driven diffusive systems, Phys. Rev. Lett. 54:2026 (1985).
H. van Beijeren, Transport properties of stochastic Lorentz models, Rev. Mod. Phys. 54:195 (1982).
C. N. Yang and C. P. Yang, Ground state energy of a Heisenberg-Ising lattice, Phys. Rev. 147:303 (1966).
G. Gallavotti, Fluctuation patterns and conditional reversibility in nonequilibrium systems, Ann. Institut H. Poincaré, in print, and chao-dyn@xyz.lanl.gov #9703007.
G. Gallavotti, private communication.
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Lebowitz, J.L., Spohn, H. A Gallavotti–Cohen-Type Symmetry in the Large Deviation Functional for Stochastic Dynamics. Journal of Statistical Physics 95, 333–365 (1999). https://doi.org/10.1023/A:1004589714161
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DOI: https://doi.org/10.1023/A:1004589714161