Two Refreshing Views of Fluctuation Theorems Through Kinematics Elements and Exponential Martingale

Journal of Statistical Physics

Volumn 143, Issue 3, 2011, Pages 543-584

  Chetrite, Raphaël ^a,b   Gupta, Shamik ^a,c  

^a WEIZMANN INSTITUTE OF SCIENCE   (Israel)

^b UNIVERSITÉ DE NICE SOPHIA ANTIPOLIS   (France)

^c UNIVERSITÉ DE LYON   (France)

Author keywords

Fluctuation Relations; Fluctuation Dissipation Theorems; Martingales; Non equilibrium Markov Process

Indexed keywords

EID: 79955842479     PISSN: 00224715     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10955-011-0184-0     Document Type: Article

References (91)

1. Agarwal, G. S.: Fluctuation-dissipation theorems for systems in non-thermal equilibrium and applications. Z. Phys. 252, 25-38 (1972).
2. Applebaum, D.: Levy Process Stochastic Calculus. Cambridge University Press, Cambridge (2004).
3. Baiesi, M., Maes, C., Wynants, B.: Fluctuations and response of nonequilibrium states. Phys. Rev. Lett. 103, 010602 (2009).
4. Baiesi, M., Maes, C., Wynants, B.: Nonequilibrium linear response for Markov dynamics, I: jump processes and overdamped diffusions. J. Stat. Phys. 137, 1094-1116 (2009).
5. Bass, R.: SDEs with jumps. Notes for Cornell Summer School. http://www. math. uconn. edu/bass/cornell. pdf (2007).
6. Baule, A., Cohen, E. G. D.: Fluctuation properties of an effective nonlinear system subject to Poisson noise. Phys. Rev. E 79, 030103 (2009).
7. Blumenthal, R. M., Getoor, R. K.: Markov Processes and Potential Theory. Academic Press, San Diego (1968).
8. Brissaud, A., Frisch, U.: Linear stochastic differential equation. J. Math. Phys. 15, 5 (1974).
9. Callen, H. B., Welton, T. A.: Irreversibility and generalized noise. Phys. Rev. 83, 34-40 (1951).
10. Chernyak, V., Chertkov, M., Jarzynski, C.: Path-integral analysis of fluctuation theorems for general Langevin processes. J. Stat. Mech. P08001 (2006).
11. Chetrite, R., Gawedzki, K.: Fluctuation relations for diffusion processes. Commun. Math. Phys. 282, 469-518 (2008).
12. Chetrite, R., Falkovich, G., Gawedzki, K.: Fluctuation relations in simple examples of nonequilibrium steady states. J. Stat. Mech. P08005 (2008).
13. Chetrite, R., Gawedzki, K.: Eulerian and Lagrangian pictures of nonequilibrium diffusions. J. Stat. Phys. 137, 5-6 (2009).
14. Chetrite, R.: Fluctuation relations for diffusion that is thermally driven by a nonstationary bath. Phys. Rev E. 051107 (2009).
15. Chetrite, R.: Thesis of ENS-Lyon (2008). Manuscript available at http://perso. ens-lyon. fr/raphael. chetrite/.
16. Chung, K. L., Walsh, J. B.: Markov Processes, Brownian Motion, and Time Symmetry, 2nd edn. Springer, Berlin (2005).
17. Cont, R., Tankov, P.: Financial Modeling with Jump Processes. Chapman & Hall, London (2003).
18. Crisanti, A., Ritort, J.: Violation of the fluctuation-dissipation theorem in glassy systems: basic notions and the numerical evidence. J. Phys. A, Math. Gen. 36, R181-R290 (2003).
19. Crooks, G. E.: Path ensembles averages in systems driven far from equilibrium. Phys. Rev. E 61, 2361-2366 (2000).
20. Cugliandolo, L. F., Kurchan, J., Parisi, G.: Off equilibrium dynamics and aging in unfrustrated systems. J. Phys. 4, 1641-1656 (1994).
21. Czernik, T., Kula, J., Luczka, J., Hanggi, P.: Thermal ratchets driven by Poissonian white shot noise. Phys. Rev. E 55, 4057-4066 (1997).
22. Darses, S., Nourdin, I.: Dynamical properties and characterization of gradient drift diffusion. Electron. Commun. Probab. 12, 390-400 (2007).
23. Davis, M. H. A.: Piecewise-deterministic Markov process: a general class of non-diffusion stochastic models. J. R. Stat. Soc. B 46, 353-388 (1984).
24. Dellacherie, C., Meyer, P. A.: Probabilités et potential. Chapitre V à VIII. In: Actualités Scientifiques et Industrielles, vol. 1385. Hermann, Paris (1980).
25. Dembo, A., Deuschel, J. D.: Markovian perturbation, response and fluctuation dissipation theorem. Ann. Inst. Henri Poincaré 46, 822-852 (2010).
26. Diaconis, P., Miclo, L.: On characterizations of Metropolis type algorithms in continuous time. ALEA Lat. Am. J. Probab. Math. Stat. 6, 199-238 (2009).
27. Diezemann, G.: Fluctuation-dissipation relations for Markov processes. Phys. Rev. E 72, 011104 (2005).
28. Doob, J. L.: Classical Potential Theory and Its Probabilistic Counterpart. Springer, New York (1984).
29. Durrett, R.: Probability, Theory and Examples, 4th edn. Cambridge University Press, Cambridge (2010).
30. Dynkin, E. B.: The initial and final behavior of trajectories of Markov processes. Russ. Math. Surv. 26, 165 (1971).
31. Dynkin, E. B.: On duality for Markov processes. In: Friddman, A., Pinsky, M. (eds.) Stochastic Analysis. Academic Press, San Diego (1978).
32. Evans, D. J., Searles, D. J.: Equilibrium microstates which generate second law violating steady states. Phys. Rev. E 50, 1645-1648 (1994).
33. Feller, W.: On the integro-differential equations of purely discontinuous Markov processes. Trans. Am. Math. Soc. 48, 488-515 (1940).
34. Follmer, H.: An entropy approach to the time reversal of diffusion process. In: Stochastic Differential Equation. Lecture Notes in Control and Information Sci., vol. 69, pp. 156-163. Springer, Berlin (1985).
35. Gallavotti, G., Cohen, E. G. D.: Dynamical ensemble in a stationary state. J. Stat. Phys. 80, 931-970 (1995).
36. Gallavotti, G.: Extension of Onsager's reciprocity to large fields and the chaotic hypothesis. Phys. Rev. Lett. 77, 4334-4337 (1996).
37. Ge, H., Jiang, D. Q.: Generalized Jarzynski's equality of inhomogeneous multidimensional diffusion processes. J. Stat. Phys. 131, 675-689 (2008).
38. Gomez-Solano, J. R., Petrosyan, A., Ciliberto, S., Chetrite, R., Gawedzki, K.: Experimental verification of a modified fluctuation-dissipation relation for a micron-sized particle in a nonequilibrium steady state. Phys. Rev. Lett. 103, 040601 (2009).
39. Gomez-Solano, J. R., Petrosyan, A., Ciliberto, S., Maes, C.: Non-equilibrium linear response of micron-sized systems. arXiv: 1006. 3196v1 (2010).
40. Hanggi, P.: Langevin description of Markovian integro-differential master equations. Z. Phys. B, Condens. Matter Quanta 36, 271-282 (1980).
41. Hanggi, P., Thomas, H.: Stochastic processes: time evolution, symmetries and linear response. Phys. Rep. 88, 207-319 (1982).
42. Harris, R. J., Schütz, G. M.: Fluctuation theorems for stochastic dynamics. J. Stat. Mech. P07020 (2007).
43. Hatano, T., Sasa, S.: Steady-state thermodynamics of Langevin systems. Phys. Rev. Lett. 86, 3463-3466 (2001).
44. Haussmann, U. G., Pardoux, E.: Time reversal of diffusions. Ann. Probab. 14, 1188-1205 (1986).
45. Ito, K., Watanabe, S.: Transformation of Markov processes by multiplicative functionals. Ann. Inst. Fourier 15, 13-30 (1965).
46. Jacob, N.: Pseudo-differential Operators and Markov Processes, vol. I. Imperial College Press, London (2001).
47. Jacob, N.: Pseudo-differential Operators and Markov Processes, vol. II. Imperial College Press, London (2002).
48. Jacob, N.: Pseudo-differential Operators and Markov Processes, vol. III. Imperial College Press, London (2005).
49. Jarzynski, C.: A nonequilibrium equality for free energy differences. Phys. Rev. Lett. 78, 2690-2693 (1997).
50. Jarzynski, C.: Equilibrium free energy differences from nonequilibrium measurements: a master equation approach. Phys. Rev. E 56, 5018 (1997).
51. Jiang, D. Q., Qian, M., Qian, M. P.: Mathematical theory of nonequilibrium steady states: on the frontier of probability and dynamical systems. In: Lecture Notes in Mathematics, vol. 1833. Springer, Berlin (2004).
52. Joachain, C.: Quantum Collision Theory. North-Holland, Amsterdam (1975).
53. Kubo, R.: The fluctuation-dissipation theorem. Rep. Prog. Phys. 29, 255-284 (1966).
54. Kubo, R., Toda, M., Hashitsume, N.: Statistical Physics II: Non-equilibrium Statistical Mechanics. Springer, Berlin (1991).
55. Klimontovich, Yu. K.: Ito, Statonovich and kinetic forms of stochastic equations. Physics A 163, 515-532 (1990).
56. Kunita, H.: Absolute continuity of Markov process and generators. Nagoya Math. J. 36, 1-26 (1969).
57. Kurchan, J.: Fluctuation theorem for stochastic dynamics. J. Phys. A, Math. Gen. 31, 3719-3729 (1998).
58. Kurchan, J.: Non-equilibrium work relations. J. Stat. Mech.: Theory Exp., P07005 (2007).
59. Lebowitz, J., Spohn, H.: A Gallavotti-Cohen type symmetry in the large deviation functional for stochastic dynamics. J. Stat. Phys. 95, 333-365 (1999).
60. Lippiello, E., Corberi, F., Zannetti, M.: Off-equilibrium generalization of the fluctuation dissipation theorem for Ising spins and measurement of the linear response function. Phys. Rev. E 71, 036104 (2005).
61. Lippiello, E., Corberi, F., Sarracinno, A., Zannetti, M.: Non-linear response and fluctuation dissipation relations. Phys. Rev. E 78, 041120 (2008).
62. Liu, F., Ou-Yang, Z. C.: Generalized integral fluctuation theorem for diffusion processes. Phys. Rev. E 79, 060107 (2009).
63. Liu, F., Luo, Y. P., Huang, M. C., Ou-Yang, Z. C.: A generalized integral fluctuation theorem for general jump processes. J. Phys. A, Math. Theor. 42, 332003 (2009).
64. Liu, F., Ou-Yang, Z. C.: Linear response theory and transient fluctuation-theorems for diffusion processes: a backward point of view. arXiv: 0912. 1917v1 (2009).
65. Luczka, J., Czernik, T., Hanggi, P.: Symmetric white noise can induce directed current in ratchets. Phys. Rev. E 56, 3968-3975 (1997).
66. Maes, C.: The fluctuation theorem as a Gibbs property. J. Stat. Phys. 95, 367-392 (1999).
67. Maes, C., Wynants, B.: On a response formula and its interpretation. Markov Process. Relat. Fields 16, 45-58 (2010).
68. Marini Bettolo Marconi, U., Puglisi, A., Rondoni, L., Vulpiani, A.: Fluctuation-dissipation: response theory in statistical physics. Phys. Rep. 461, 111-195 (2008).
69. Mayer, P., Leonard, S., Berthier, L., Garrahan, J. P., Sollich, P.: Activated aging dynamics and negative fluctuation-dissipation ratios. Phys. Rev. Lett. 96, 030602 (2006).
70. Merton, R.: Option pricing when underlying stock returns are discontinuous. J. Financ. Econ. 3, 125-144 (1976).
71. Nelson, E.: Dynamical Theories of Brownian Motion, 2nd edn. Princeton University Press, Princeton (2001).
72. Nelson, E.: Quantum Fluctuations, Princeton Series in Physics. Princeton University Press, Princeton (1985).
73. Onsager, L.: Reciprocal relations in irreversible processes I. Phys. Rev. 37, 405-426 (1931).
74. Onsager, L.: Reciprocal relations in irreversible processes II. Phys. Rev. 38, 2265-2279 (1931).
75. Oono, Y., Paniconi, M.: Steady state thermodynamics. Prog. Theor. Phys. Suppl. 130, 29-44 (1998).
76. Palmowski, Z., Rolski, T.: A technique for exponential change of measure for Markov processes. Bernoulli 8(6), 767-785 (2002).
77. Porporato, A., D'Odorico, P.: Phase transitions driven by state-dependent Poisson noise. Phys. Rev. Lett. 92, 110601 (2004).
78. Prost, J., Joanny, J. F., Parrondo, J. M. R.: Generalized fluctuation-dissipation theorem for steady state systems. Phys. Rev. Lett. 103, 090601 (2009).
79. Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion, 3rd edn. Springer, Berlin (1999).
80. Risken, H.: The Fokker Planck Equation, 2nd edn. Springer, Berlin-Heidelberg (1989).
81. Sancho, J. M., San Miguel, M., Pesquera, L., Rodriguez, M. A.: Positivity requirements on fluctuating parameters. Physica A 142, 532-547 (1987).
82. Seifert, U.: Stochastic thermodynamics: Principles and perspectives. Eur. Phys. J. B 64, 423-432 (2008).
83. Seifert, U., Speck, T.: Fluctuation-dissipation theorem in nonequilibrium steady states. Europhys. Lett. 89, 10007 (2010).
84. Sekimoto, K.: Stochastic energetics. In: Lecture Notes in Physics, vol. 799. Springer, Berlin (2010).
85. Speck, T., Seifert, U.: Restoring a fluctuation-dissipation theorem in a nonequilibrium steady state. Europhys. Lett. 74, 391-396 (2006).
86. Speck, T.: Driven soft matter: entropy production and the fluctuation-dissipation theorem. arXiv: 1004. 1621 (2010).
87. Stroock, D., Varadhan, S. R. S.: Multidimensional Diffusion Processes. Springer, Berlin (1979).
88. Stroock, D.: Diffusion processes associated with Lévy generators. Probab. Theory Relat. Fields 32, 209-244 (1975).
89. Touchette, H., Cohen, E. G. D.: Fluctuation relation for a Levy particle. Phys. Rev. E 76, 020101 (2007).
90. van Kampen, N. G.: Processes with delta-correlated cumulants. Physica A 102, 489-495 (1980).
91. Zimmer, M. F.: Fluctuations in nonequilibrium systems and broken supersymmetry. J. Stat. Phys. 73, 751-764 (1993).