The large deviation approach to statistical mechanics

Physics Reports

Volumn 478, Issue 1-3, 2009, Pages 1-69

  Touchette, Hugo ^a  

^a QUEEN MARY UNIVERSITY OF LONDON   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 67650917097     PISSN: 03701573     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.physrep.2009.05.002     Document Type: Review

References (292)

1. Cramér H. Sur un nouveau théorème limite dans la théorie des probabilités. Colloque consacré à la théorie des probabilités vol. 3 (1938), Hermann, Paris 2-23
2. Donsker M.D., and Varadhan S.R.S. Asymptotic evaluation of certain Markov process expectations for large time. I. Comm. Pure Appl. Math. 28 (1975) 1-47
3. Donsker M.D., and Varadhan S.R.S. Asymptotic evaluation of certain Markov process expectations for large time. II. Comm. Pure Appl. Math. 28 (1975) 279-301
4. Donsker M.D., and Varadhan S.R.S. Asymptotic evaluation of certain Markov process expectations for large time. III. Comm. Pure Appl. Math. 29 4 (1976) 389-461
5. Donsker M.D., and Varadhan S.R.S. Asymptotic evaluation of certain Markov process expectations for large time. IV. Comm. Pure Appl. Math. 36 2 (1983) 183-212
6. Freidlin M.I., and Wentzell A.D. Random Perturbations of Dynamical Systems. Grundlehren der Mathematischen Wissenschaften vol. 260 (1984), Springer-Verlag, New York
7. Ellis R.S. The theory of large deviations: From Boltzmann's 1877 calculation to equilibrium macrostates in 2D turbulence. Physica D 133 (1999) 106-136
8. Ellis R.S. Entropy, Large Deviations, and Statistical Mechanics (1985), Springer, New York
9. Oono Y. Large deviation and statistical physics. Progr. Theoret. Phys. Suppl. 99 (1989) 165-205
10. Ellis R.S. An overview of the theory of large deviations and applications to statistical mechanics. Scand. Actuar. J. 1 (1995) 97-142
11. Lanford O.E. Entropy and equilibrium states in classical statistical mechanics. In: Lenard A. (Ed). Statistical Mechanics and Mathematical Problems. Lecture Notes in Physics vol. 20 (1973), Springer, Berlin 1-113
12. Eyink G.L. Dissipation and large thermodynamic fluctuations. J. Statist. Phys. 61 3 (1990) 533-572
13. Dembo A., and Zeitouni O. Large Deviations Techniques and Applications. 2nd ed. (1998), Springer, New York
14. Cover T.M., and Thomas J.A. Elements of Information Theory (1991), John Wiley, New York
15. Dupuis P., and Ellis R.S. A Weak Convergence Approach to the Theory of Large Deviations. Wiley Series in Probability and Statistics (1997), John Wiley, New York
16. Capiński M., and Kopp E. Measure, Integral and Probability. Springer Undergraduate Mathematics Series (2005), Springer, New York
17. Gärtner J. On large deviations from the invariant measure. Theory Probab. Appl. 22 (1977) 24-39
18. Ellis R.S. Large deviations for a general class of random vectors. Ann. Probab. 12 1 (1984) 1-12
19. Rockafellar R.T. Convex Analysis (1970), Princeton University Press, Princeton
20. Bender C.M., and Orszag S.A. Advanced Mathematical Methods for Scientists and Engineers (1978), McGraw-Hill, New York
21. van Tiel J. Convex Analysis: An Introductory Text (1984), John Wiley, New York
22. Varadhan S.R.S. Asymptotic probabilities and differential equations. Comm. Pure Appl. Math. 19 (1966) 261-286
23. O'Connell N. From laws of large numbers to large deviation principles. Markov Process. Related Fields 3 4 (1997)
24. O'Connell N. A large deviations heuristic made precise. Math. Proc. Cambridge Philos. Soc. 128 (2000) 561-569
25. Martin-Löf A. A Laplace approximation for sums of independent random variables. Z. Wahrscheinlichkeitstheor. Verwandte Geb. 59 1 (1982) 101-115
26. Bryc W. A remark on the connection between the large deviation principle and the central limit theorem. Statist. Probab. Lett. 18 4 (1993) 253-256
27. d. Ann. Probab. 11 1 (1983) 158-167
28. Bucklew J.A. Large Deviation Techniques in Decision, Simulation and Estimation. Wiley Series in Probability and Mathematical Statistics (1990), Wiley Interscience, New York
29. Sanov I.N. On the probability of large deviations of random variables. Select. Transl. Math. Statist. and Probability vol. 1 (1961), Inst. Math. Statist. and Amer. Math. Soc., Providence, R.I. 213-244
30. Boltzmann L. Über die Beziehung zwischen dem zweiten Hauptsatze der mechanischen Wärmetheorie und der Wahrscheinlichkeitsrechnung respecktive den Sätzen über das Wärmegleichgewicht (On the relationship between the second law of the mechanical theory of heat and the probability calculus). Wiener Berichte 2 76 (1877) 373-435
31. Bahadur R.R., and Zabell S.L. Large deviations of the sample mean in general vector spaces. Ann. Probab. 7 4 (1979) 587-621
32. Plachky D., and Steinebach J. A theorem about probabilities of large deviations with an application to queuing theory. Period. Math. Hungar. 6 (1975) 343-345
33. Daniels H.E. Saddlepoint approximations in statistics. Ann. Math. Stat. 25 (1954) 631-650
34. Barndorff-Nielsen O.E., and Cox D.R. Asymptotic Techniques for Use in Statistics. Monographs on Statistics and Applied Probability (1989), Chapman and Hall, London
35. Butler R.W. Saddlepoint Approximations with Applications (2007), Cambridge University Press, Cambridge
36. Amann A., and Atmanspacher H. Introductory remarks on large deviation statistics. J. Sci. Exploration 13 (1999) 639-664
37. J.T. Lewis, R. Russell, An introduction to large deviations for teletraffic engineers, DIAS Report, 1996
38. Varadhan S.R.S. Large deviations and entropy. In: Greven A., Keller G., and Warnecke G. (Eds). Entropy. Princeton Ser. Appl. Math. (2003), Princeton University Press, Princeton 199-214 (Chapter 9)
39. Deuschel J.D., and Stroock D.W. Large Deviations (1989), Academic Press, Boston
40. den Hollander F. Large Deviations. Fields Institute Monograph (2000), Amer. Math. Soc., Providence, R.I.
41. Uchaikin V.V., and Zolotarev V.M. Chance and Stability: Stable Distributions and their Applications (1999), VSP, Utrecht
42. Nagaev A.V. Asymptotic properties of stable densities and the asymmetric large deviation problems. Statist. Probab. Lett. 61 4 (2003) 429-438
43. Nagaev A.V. Cramér's large deviations when the extreme conjugate distribution is heavy-tailed. Theory Probab. Appl. 43 (1999) 405-421
44. Ney P., and Nummelin E. Markov additive processes I: Eigenvalue properties and limit theorems. Ann. Probab. 15 (1987) 561-592
45. Ney P., and Nummelin E. Markov additive processes II: Large deviations. Ann. Probab. 15 (1987) 593-609
46. Dinwoodie I.H., and Zabell S.L. Large deviations for exchangeable random vectors. Ann. Probab. 20 (1992) 1147-1166
47. Dinwoodie I.H. Identifying a large deviation rate function. Ann. Probab. 21 (1993) 216-231
48. Ioffe D. Two examples in the theory of large deviations. Statist. Probab. Lett. 18 (1993) 297-300
49. Lecomte V., Appert-Rolland C., and van Wijland F. Chaotic properties of systems with Markov dynamics. Phys. Rev. Lett. 95 1 (2005) 010601
50. Lecomte V., Appert-Rolland C., and van Wijland F. Thermodynamic formalism for systems with Markov dynamics. J. Statist. Phys. (2007)
51. Beck C., and Schlögl F. Thermodynamics of Chaotic Systems: An Introduction (1993), Cambridge University Press, Cambridge
52. Gaspard P. Chaos, Scattering and Statistical Mechanics. Cambridge Nonlinear Science Series vol. 9 (1998), Cambridge University Press, Cambridge
53. Ruelle D. Statistical Mechanics: Rigorous Results (1969), W.A. Benjamin, Amsterdam
54. Lewis J.T. The large deviation principle in statistical mechanics: An expository account. In: Truman A., and Davies I.M. (Eds). Stochastic Mechanics and Stochastic Processes. Lecture Notes in Mathematics vol. 1325 (1988), Springer, New York 141-155
55. Lewis J.T. The large deviation principle in statistical mechanics. Mark Kac Seminar on Probability and Physics vol. 17 (1988), Math. Centrum Centrum Wisk. Inform., Amsterdam 85-102
56. Lewis J.T. Large deviations and statistical mechanics. Mathematical Methods in Statistical Mechanics. Leuven Notes Math. Theoret. Phys. Ser. A Math. Phys. vol. 1 (1989), Leuven Univ. Press, Leuven 77-90
57. Lewis J.T., Pfister C.-E., and Sullivan W.G. Large deviations and the thermodynamic formalism: A new proof of the equivalence of ensembles. In: Fannes M., Maes C., and Verbeure A. (Eds). On Three Levels (1994), Plenum Press, New York
58. Lewis J.T., Pfister C.-E., and Sullivan W.G. Entropy, concentration of probability and conditional limit theorem. Markov Process. Related Fields 1 (1995) 319-386
59. Lewis J.T., and Pfister C.-E. Thermodynamic probability theory: Some aspects of large deviations. Russ. Math. Surveys 50 2 (1995) 279-317
60. Pfister C.-E. Large deviations and phase separation in the two-dimensional Ising model. Helv. Phys. Acta 64 7 (1991) 953-1054
61. Balian R. From Microphysics to Macrophysics: Methods and Applications of Statistical Physics, vol. I (1991), Springer, Berlin
62. Ellis R.S., Haven K., and Turkington B. Large deviation principles and complete equivalence and nonequivalence results for pure and mixed ensembles. J. Statist. Phys. 101 (2000) 999-1064
63. Einstein A. The theory of opalescence of homogeneous fluids and liquid mixtures near the critical state. In: Stachel J. (Ed). The Collected Papers of Albert Einstein vol. 3 (1987), Princeton University Press, Princeton 231-249
64. Jaynes E.T. Information theory and statistical mechanics. Phys. Rev. 106 (1957) 620-630
65. Jaynes E.T. Probability Theory: The Logic of Science Vol. I (2003), Cambridge University Press, Cambridge
66. Lehtonen T., and Nummelin E. Level I theory of large deviations in the ideal gas. Internat. J. Theoret. Phys. 29 (1990) 621-635
67. Eisele T., and Ellis R.S. Multiple phase transitions in the generalized Curie-Weiss model. J. Statist. Phys. 52 1 (1988) 161-202
68. Orey S. Large deviations for the empirical field of Curie-Weiss models. Stochastics 25 (1988) 3-14
69. Ellis R.S., and Wang K. Limit theorems for the empirical vector of the Curie-Weiss-Potts model. Stochastic Process. Appl. 35 (1990) 59-79
70. Costeniuc M., Ellis R.S., and Touchette H. Complete analysis of phase transitions and ensemble equivalence for the Curie-Weiss-Potts model. J. Math. Phys. 46 (2005) 063301
71. Barré J., Mukamel D., and Ruffo S. Inequivalence of ensembles in a system with long-range interactions. Phys. Rev. Lett. 87 (2001) 030601
72. Ellis R.S., Touchette H., and Turkington B. Thermodynamic versus statistical nonequivalence of ensembles for the mean-field Blume-Emery-Griffiths model. Physica A 335 (2004) 518-538
73. Ellis R.S., Otto P., and Touchette H. Analysis of phase transitions in the mean-field Blume-Emery-Griffiths model. Ann. Appl. Probab. 15 (2005) 2203-2254
74. Barré J., Bouchet F., Dauxois T., and Ruffo S. Large deviation techniques applied to systems with long-range interactions. J. Statist. Phys. 119 3 (2005) 677-713
75. Kastner M., and Schnetz O. On the mean-field spherical model. J. Statist. Phys. 122 6 (2006) 1195-1214
76. Casetti L., and Kastner M. Partial equivalence of statistical ensembles and kinetic energy. Physica A 384 2 (2007) 318-334
77. 4 model: Entropy, analyticity, and configuration space topology. Phys. Rev. E 72 (2005) 056134
78. 4-model. Eur. Phys. J. B 50 (2006) 311-314
79. 4 spin model. Physica A 385 1 (2007) 233-248
80. In: Dauxois T., Ruffo S., Arimondo E., and Wilkens M. (Eds). Dynamics and Thermodynamics of Systems with Long Range Interactions, vol. 602 (2002), Springer, New York
81. Bouchet F., and Barré J. Classification of phase transitions and ensemble inequivalence in systems with long range interactions. J. Statist. Phys. 118 5 (2005) 1073-1105
82. In: Campa A., Giansanti A., Morigi G., and Labini F.S. (Eds). Dynamics and Thermodynamics of Systems with Long-range Interactions: Theory and Experiments (2008), AIP, Melville, NY
83. A. Campa, T. Dauxois, S. Ruffo, Statistical mechanics and dynamics of solvable models with long-range interactions, 2008
84. Ellis R.S., Haven K., and Turkington B. Nonequivalent statistical equilibrium ensembles and refined stability theorems for most probable flows. Nonlinearity 15 (2002) 239-255
85. Pfister C.-E. Thermodynamical aspects of classical lattice systems. In: Sidoravicius V. (Ed). In and Out of Equilibrium, Probability with a Physics Flavor (2002), Birkhäuser, Boston 393-472
86. Kastner M. Existence and order of the phase transition of the Ising model with fixed magnetization. J. Statist. Phys. 109 1 (2002) 133-142
87. Lewis J.T., Pfister C.-E., and Sullivan G.W. The equivalence of ensembles for lattice systems: Some examples and a counterexample. J. Statist. Phys. 77 (1994) 397-419
88. Onsager L. Crystal statistics. I. A two-dimensional model with an order-disorder transition. Phys. Rev. 65 3-4 (1944) 117-149
89. Ioffe D. Large deviations for the 2D Ising model: A lower bound without cluster expansions. J. Statist. Phys. 74 1 (1994) 411-432
90. Landau L.D., and Lifshitz E.M. Statistical Physics. 3rd ed. Landau and Lifshitz Course of Theoretical Physics vol. 5 (1991), Butterworth Heinemann, Oxford
91. Touchette H. Comment on First-order phase transition: Equivalence between bimodalities and the Yang-Lee theorem. Physica A 359 (2006) 375-379
92. Maxwell J.C. On the dynamical evidence of the molecular constitution of bodies. Nature II (1875) 357
93. Huang K. Statistical Mechanics (1987), Wiley, New York
94. Ispolatov I., and Cohen E.G.D. On first-order phase transitions in microcanonical and canonical non-extensive systems. Physica A 295 (2000) 475-487
95. Kiessling M.K.-H., and Neukirch T. Negative specific heat of a magnetically self-confined plasma torus. Proc. Natl. Acad. Sci. USA 100 4 (2003) 1510-1514
96. Kiessling M.K.-H., and Lebowitz J. The micro-canonical point vortex ensemble: Beyond equivalence. Lett. Math. Phys. 42 (1997) 43-56
97. Chavanis P.-H. Phase transitions in self-gravitating systems. Internat. J. Modern Phys. B 20 22 (2006) 3113-3198
98. Lynden-Bell D. Negative specific heat in astronomy, physics and chemistry. Physica A 263 (1999) 293-304
99. Lebowitz J.L. Boltzmann's entropy and time's arrow. Phys. Today 46 (1993) 32-38
100. Eyink G.L., and Spohn H. Negative-temperature states and large-scale, long-lived vortices in two-dimensional turbulence. J. Statist. Phys. 70 (1993) 833-886
101. Touchette H. Simple spin models with non-concave entropies. Amer. J. Phys. 76 (2008) 26-30
102. Touchette H., Ellis R.S., and Turkington B. An introduction to the thermodynamic and macrostate levels of nonequivalent ensembles. Physica A 340 (2004) 138-146
103. Ruelle D. Thermodynamic Formalism. 2nd ed. (2004), Cambridge University Press, Cambridge
104. Griffiths R.B. Microcanonical ensemble in quantum statistical mechanics. J. Math. Phys. 6 10 (1965) 1447-1461
105. Galgani L., Manzoni L., and Scotti A. Asymptotic equivalence of equilibrium ensembles of classical statistical mechanics. J. Math. Phys. 12 6 (1971) 933-935
106. Gallavotti G. Statistical Mechanics: A Short Treatise (1999), Springer, New York
107. van Hove L. Quelques propriétés générales de l'intégrale de configuration d'un système de particules avec interaction. Physica 15 11-12 (1949) 951-961
108. van Hove L. Sur l'intégrale de configuration pour les systèmes de particules à une dimension. Physica 16 2 (1950) 137-143
109. Ruelle D. Classical statistical mechanics of a system of particles. Helv. Phys. Acta 36 (1963) 183-197
110. Fisher M.E. The free energy of a macroscopic system. Arch. Ration. Mech. Anal. 17 5 (1964) 377-410
111. Griffiths R.B. A proof that the free energy of a spin system is extensive. J. Math. Phys. 5 9 (1964) 1215-1222
112. Griffiths R.B. Rigorous results and theorems. In: Domb C., and Green M.S. (Eds). Phase Transitions and Critical Phenomena vol. 1 (1972), Academic Press, London 7-109
113. Derrida B. Non-equilibrium steady states: Fluctuations and large deviations of the density and of the current. J. Stat. Mech. 2007 07 (2007) P07023
114. Gardiner C.W. Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences. 2nd ed. Springer Series in Synergetics vol. 13 (1985), Springer, New York
115. Wentzell A.D., and Freidlin M.I. On small random perturbations of dynamical systems. Russ. Math. Surveys 25 (1970) 1-55
116. Graham R. Statistical theory of instabilities in stationary nonequilibrium systems with applications to lasers and nonlinear optics. In: Höhler G. (Ed). Quantum Statistics in Optics and Solid-State Physics. Springer Tracts in Modern Physics vol. 66 (1973), Springer, Berlin 1-97
117. Graham R., and Tél T. Weak-noise limit of Fokker-Planck models and nondifferentiable potentials for dissipative dynamical systems. Phys. Rev. A 31 2 (1985) 1109-1122
118. Dykman M.I., and Krivoglaz M.A. Theory of fluctuational transitions between stable states of nonlinear oscillators. Sov. Phys. JETP 50 1 (1979) 30-37
119. Onsager L., and Machlup S. Fluctuations and irreversible processes. Phys. Rev. 91 6 (1953) 1505-1512
120. Falkoff D. Integral over path formulation of statistical theory of irreversible processes. Progr. Theoret. Phys. 16 5 (1956) 530-532
121. Falkoff D. Statistical theory of irrversible processes: Part I. Intergral over fluctuation path formulation. Ann. Phys. (N.Y.) 4 3 (1958) 325-346
122. Stratonovich R.L. Some Markov methods in the theory of stochastic processes in nonlinear dynamical systems. In: Moss F., and McClintock P.V.E. (Eds). Noise in Nonlinear Dynamical Systems vol. 1 (1989), Cambridge University Press, Cambridge 16-72
123. Dürr D., and Bach A. The Onsager-Machlup function as Lagrangian for the most probable path of a diffusion process. Comm. Math. Phys. 60 2 (1978) 153-170
124. Horsthemke W., and Bach A. Onsager-Machlup function for one dimensional nonlinear diffusion processes. Z. Phys. B (Condens. Matter) 22 2 (1975) 189-192
125. Hunt K.L.C., and Ross J. Path integral solutions of stochastic equations for nonlinear irreversible processes: The uniqueness of the thermodynamic lagrangian. J. Chem. Phys. 75 2 (1981) 976-984
126. Bray A.J., McKane A.J., and Newman T.J. Path integrals and non-Markov processes. II. Escape rates and stationary distributions in the weak-noise limit. Phys. Rev. A 41 2 (1990) 657-667
127. McKane A.J., Luckock H.C., and Bray A.J. Path integrals and non-Markov processes. I. General formalism. Phys. Rev. A 41 2 (1990) 644-656
128. Wio H.S., Colet P., Miguel M.S., Pesquera L., and Rodríguez M.A. Path-integral formulation for stochastic processes driven by colored noise. Phys. Rev. A 40 12 (1989) 7312-7324
129. Dykman M.I., and Lindenberg K. Fluctuations in nonlinear systems driven by colored noise. In: Weiss G.H. (Ed). Contemporary Problems in Statistical Physics (1994), SIAM, Philadelphia 41-101
130. Einchcomb S.J.B., and McKane A.J. Use of Hamiltonian mechanics in systems driven by colored noise. Phys. Rev. E 51 4 (1995) 2974-2981
131. In: Moss F., and McClintock P.V.E. (Eds). Noise in Nonlinear Dynamical Systems, vol. 1 (1989), Cambridge University Press, Cambridge
132. In: Moss F., and McClintock P.V.E. (Eds). Noise in Nonlinear Dynamical Systems, vol. 2 (1989), Cambridge University Press, Cambridge
133. Kleinert H. Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets (2004), World Scientific, Singapore
134. Wiegel F.W. Introduction to Path-integral Methods in Physics and Polymer Science (1986), World Scientific, Singapore
135. Schilder M. Some asymptotic formulae for Wiener integrals. Trans. Amer. Math. Soc. 125 (1966) 63-85
136. Roy R.V. Noise perturbation of nonlinear dynamical systems. In: Cheng A.H.-D., and Yang C.Y. (Eds). Computational Stochastic Mechanics (1993), Elsevier, Amsterdam 125-148
137. Roy R.V. Large deviation theory, weak-noise asymptotics, and first-passage problems: Review and results. In: Lemaire M., Favre J.-L., and Mebarki A. (Eds). Applications of Statistics and Probability (1995), A.A. Balkema, Rotterdam 1129-1135
138. Dykman M.I., Mori E., Ross J., and Hunt P.M. Large fluctuations and optimal paths in chemical kinetics. J. Chem. Phys. 100 8 (1994) 5735-5750
139. Bray A.J., and McKane A.J. Instanton calculation of the escape rate for activation over a potential barrier driven by colored noise. Phys. Rev. Lett. 62 5 (1989) 493-496
140. Bertini L., Sole A.D., Gabrielli D., Jona-Lasinio G., and Landim C. Macroscopic fluctuation theory for stationary non-equilibrium states. J. Statist. Phys. 107 3 (2002) 635-675
141. van Kampen N.G. Stochastic Processes in Physics and Chemistry (1992), North-Holland, Amsterdam
142. Kautz R.L. Thermally induced escape: The principle of minimum available noise energy. Phys. Rev. A 38 4 (1988) 2066-2080
143. Kautz R.L. Activation energy for thermally induced escape from a basin of attraction. Phys. Lett. A 125 6-7 (1987) 315-319
144. Graham R., and Tél T. Nonequilibrium potential for coexisting attractors. Phys. Rev. A 33 2 (1986) 1322-1337
145. Jauslin H.R. Nondifferentiable potentials for nonequilibrium steady states. Physica A 144 1 (1987) 179-191
146. Dykman M.I., McClintock P.V.E., Smelyanski V.N., Stein N.D., and Stocks N.G. Optimal paths and the prehistory problem for large fluctuations in noise-driven systems. Phys. Rev. Lett. 68 18 (1992) 2718-2721
147. Dykman M.I., Luchinsky D.G., McClintock P.V.E., and Smelyanskiy V.N. Corrals and critical behavior of the distribution of fluctuational paths. Phys. Rev. Lett. 77 26 (1996) 5229-5232
148. Luchinsky D.G., and McClintock P.V.E. Irreversibility of classical fluctuations studied in analogue electrical circuits. Nature 389 (1997) 463-466
149. Luchinsky D.G., Maier R.S., Mannella R., McClintock P.V.E., and Stein D.L. Experiments on critical phenomena in a noisy exit problem. Phys. Rev. Lett. 79 17 (1997) 3109-3112
150. Luchinsky D.G., McClintock P.V.E., and Dykman M.I. Analogue studies of nonlinear systems. Rep. Progr. Phys. 61 8 (1998) 889-997
151. Olivieri E. Metastability and entropy. In: Greven A., Keller G., and Warnecke G. (Eds). Entropy. Princeton Ser. Appl. Math. (2003), Princeton University Press, Princeton 233-250 (Chapter 11)
152. Olivieri E., and Vares M.E. Large Deviations and Metastability. Encyclopedia of Mathematics and Its Applications vol. 100 (2005), Cambridge University Press, Cambridge
153. Hasegawa H. Variational principle for non-equilibrium states and the Onsager-Machlup formula. Progr. Theoret. Phys. 56 1 (1976) 44-60
154. Taniguchi T., and Cohen E.G.D. Onsager-Machlup theory for nonequilibrium steady states and fluctuation theorems. J. Statist. Phys. 126 1 (2007) 1-41
155. Taniguchi T., and Cohen E.G.D. Nonequilibrium steady state thermodynamics and fluctuations for stochastic systems. J. Statist. Phys. 130 4 (2008) 633-667
156. Graham R. Onset of cooperative behavior in nonequilibrium steady states. In: Nicolis G., Dewel G., and Turner J.W. (Eds). Order and Fluctuations in Equilibrium and Nonequilibrium Statistical Mechanics (1981), Wiley, New York
157. Paniconi M., and Oono Y. Phenomenological framework for fluctuations around steady state. Phys. Rev. E 55 1 (1997) 176-188
158. Oono Y., and Paniconi M. Steady state thermodynamics. Progr. Theoret. Phys. Suppl. 130 (1998) 29-44
159. Suarez A., Ross J., Peng B., Hunt K.L.C., and Hunt P.M. Thermodynamic and stochastic theory of nonequilibrium systems: A Lagrangian approach to fluctuations and relation to excess work. J. Chem. Phys. 102 11 (1995) 4563-4573
160. Falkovich G., Gawedzki K., and Vergassola M. Particles and fields in fluid turbulence. Rev. Modern Phys. 73 4 (2001) 913-975
161. Chetrite R., Delannoyand J.-Y., and Gawedzki K. Kraichnan flow in a square: An example of integrable chaos. J. Statist. Phys. 126 6 (2007) 1165-1200
162. Gourcy M. A large deviation principle for 2D stochastic Navier-Stokes equation. Stochastic Process. Appl. 117 7 (2007) 904-927
163. Spohn H. Large Scale Dynamics of Interacting Particles (1991), Springer Verlag, Heidelberg
164. Kipnis C., and Landim C. Scaling Limits of Interacting Particle Systems. Grundlheren der mathematischen Wissenschaften vol. 320 (1999), Springer-Verlag, Berlin
165. Majumdar S.N., and Bray A.J. Large-deviation functions for nonlinear functionals of a Gaussian stationary Markov process. Phys. Rev. E 65 5 (2002) 051112
166. van Zon R., and Cohen E.G.D. Stationary and transient work-fluctuation theorems for a dragged Brownian particle. Phys. Rev. E 67 (2003) 046102
167. Touchette H., and Cohen E.G.D. Fluctuation relation for a Lévy particle. Phys. Rev. E 76 2 (2007) 020101
168. C. Maes, K. Netočný, B. Shergelashvili, A selection of nonequilibrium issues, in: Lecture notes from the 5th Prague Summer School on Mathematical Statistical Mechanics, 2006
169. Harris R.J., and Schütz G.M. Fluctuation theorems for stochastic dynamics. J. Stat. Mech. 2007 07 (2007) P07020
170. Kurchan J. Non-equilibrium work relations. J. Stat. Mech. 2007 07 (2007) P07005
171. Marconi U.M.B., Puglisi A., Rondoni L., and Vulpiani A. Fluctuation-dissipation: Response theory in statistical physics. Phys. Rep. 461 4-6 (2008) 111-195
172. Ritort F. Work fluctuations, transient violations of the second law and free-energy recovery methods: Perspectives in theory and experiments. In: Dalibard J., Duplantier B., and Rivasseau V. (Eds). Poincaré Seminar 2003. Progress in Mathematical Physics vol. 38 (2003), Birkhäuser Verlag, Basel 192-229
173. Evans D.J., and Searles D.J. The fluctuation theorem. Adv. Phys. 51 7 (2002) 1529-1585
174. Bochkov G.N., and Kuzovlev Y.E. Nonlinear fluctuation-dissipation relations and stochastic models in nonequilibrium thermodynamics : I. Generalized fluctuation-dissipation theorem. Physica A 106 3 (1981) 443-479
175. Maes C., and Netočný K. Time-reversal and entropy. J. Statist. Phys. 110 1 (2003) 269-310
176. Gallavotti G., and Cohen E.G.D. Dynamical ensembles in nonequilibrium statistical mechanics. Phys. Rev. Lett. 74 14 (1995) 2694-2697
177. Gallavotti G., and Cohen E.G.D. Dynamical ensembles in stationary states. J. Statist. Phys. 80 5 (1995) 931-970
178. Evans D.J., Cohen E.G.D., and Morriss G.P. Probability of second law violations in shearing steady states. Phys. Rev. Lett. 71 15 (1993) 2401-2404
179. Kurchan J. Fluctuation theorem for stochastic dynamics. J. Phys. A: Math. Gen. 31 (1998) 3719-3729
180. Lebowitz J.L., and Spohn H. A Gallavotti-Cohen-type symmetry in the large deviation functional for stochastic dynamics. J. Statist. Phys. 95 (1999) 333-365
181. Maes C. The fluctuation Theorem as a Gibbs property. J. Statist. Phys. 95 (1999) 367-392
182. Wang G.M., Sevick E.M., Mittag E., Searles D.J., and Evans D.J. Experimental demonstration of violations of the second law of thermodynamics for small systems and short time scales. Phys. Rev. Lett. 89 5 (2002) 050601
183. Andrieux D., Gaspard P., Ciliberto S., Garnier N., Joubaud S., and Petrosyan A. Entropy production and time asymmetry in nonequilibrium fluctuations. Phys. Rev. Lett. 98 15 (2007) 150601
184. van Zon R., Ciliberto S., and Cohen E.G.D. Power and heat fluctuation theorems for electric circuits. Phys. Rev. Lett. 92 13 (2004) 130601
185. Garnier N., and Ciliberto S. Nonequilibrium fluctuations in a resistor. Phys. Rev. E 71 6 (2005) 060101
186. Aumaître S., Fauve S., McNamara S., and Poggi P. Power injected in dissipative systems and the fluctuation theorem. Eur. Phys. J. B 19 3 (2001) 449-460
187. Feitosa K., and Menon N. Fluidized granular medium as an instance of the fluctuation theorem. Phys. Rev. Lett. 92 16 (2004) 164301
188. Puglisi A., Visco P., Barrat A., Trizac E., and van Wijland F. Fluctuations of internal energy flow in a vibrated granular gas. Phys. Rev. Lett. 95 11 (2005) 110202
189. Visco P., Puglisi A., Barrat A., Trizac E., and van Wijland F. Injected power and entropy flow in a heated granular gas. Europhys. Lett. 72 1 (2005) 55-61
190. Visco P., Puglisi A., Barrat A., Trizac E., and van Wijland F. Fluctuations of power injection in randomly driven granular gases. J. Statist. Phys. 125 (2006) 533-568
191. Ciliberto S., and Laroche S. An experimental test of the Gallavotti-Cohen fluctuation theorem. J. Phys. IV (France) 8 6 (1998) 215-219
192. Ciliberto S., Garnier N., Hernandez S., Lacpatia C., Pinton J.-F., and Chavarria G.R. Experimental test of the Gallavotti-Cohen fluctuation theorem in turbulent flows. Physica A 340 1-3 (2004) 240-250
193. Cleuren B., Van den Broeck C., and Kawai R. Fluctuation theorem for the effusion of an ideal gas. Phys. Rev. E 74 (2006) 021117
194. Gaspard P. Time-reversed dynamical entropy and irreversibility in Markovian random processes. J. Statist. Phys. 117 3 (2004) 599-615
195. Chetrite R., and Gawedzki K. Fluctuation relations for diffusion processes. Comm. Math. Phys. 282 2 (2008) 469-518
196. Chetrite R., Falkovich G., and Gawedzki K. Fluctuation relations in simple examples of non-equilibrium steady states. J. Stat. Mech. 2008 08 (2008) P08005
197. Maes C., and Netočný K. Minimum entropy production principle from a dynamical fluctuation law. J. Math. Phys. 48 (2007) 053306
198. Jiang D.-Q., Qian M., and Zhang F.-X. Entropy production fluctuations of finite Markov chains. J. Math. Phys. A: Math. Gen. 44 9 (2003) 4176-4188
199. Maes C., Redig F., and Van Moffaert A. On the definition of entropy production, via examples. J. Math. Phys. 41 3 (2000) 1528-1554
200. Maes C. On the origin and the use of fluctuation relations for the entropy. Sem. Poincaré 2 (2003) 29-62
201. Seifert U. Entropy production along a stochastic trajectory and an integral fluctuation theorem. Phys. Rev. Lett. 95 4 (2005) 040602
202. Imparato A., and Peliti L. Fluctuation relations for a driven Brownian particle. Phys. Rev. E 74 2 (2006) 026106
203. Gallavotti G. Breakdown and regeneration of time reversal symmetry in nonequilibrium statistical mechanics. Physica D 112 1-2 (1998) 250-257
204. Gallavotti G. Heat and fluctuations from order to chaos. Eur. Phys. J. B 61 1 (2008) 1-24
205. van Zon R., and Cohen E.G.D. Extension of the fluctuation theorem. Phys. Rev. Lett. 91 (2003) 110601
206. van Zon R., and Cohen E.G.D. Extended heat-fluctuation theorems for a system with deterministic and stochastic forces. Phys. Rev. E 69 (2004) 056121
207. Farago J. Injected power fluctuations in Langevin equation. J. Statist. Phys. 107 (2002) 781-803
208. Puglisi A., Rondoni L., and Vulpiani A. Relevance of initial and final conditions for the fluctuation relation in Markov processes. J. Stat. Mech. 2006 08 (2006) P08010
209. Visco P. Work fluctuations for a Brownian particle between two thermostats. J. Stat. Mech. 2006 06 (2006) P06006
210. Bonetto F., Gallavotti G., Giuliani A., and Zamponi F. Chaotic hypothesis, fluctuation theorem and singularities. J. Phys. Stat. 123 (2006) 39-54
211. Harris R.J., Rákos A., and Schütz G.M. Breakdown of Gallavotti-Cohen symmetry for stochastic dynamics. Europhys. Lett. 75 (2006) 227-233
212. Rákos A., and Harris R.J. On the range of validity of the fluctuation theorem for stochastic Markovian dynamics. J. Stat. Mech. 2008 05 (2008) P05005
213. Liggett T.M. Interacting Particle Systems. Classics in Mathematics (2004), Springer, New York
214. Bertini L., De Sole A., Gabrielli D., Jona-Lasinio G., and Landim C. Minimum dissipation principle in stationary non-equilibrium states. J. Statist. Phys. 116 1 (2004) 831-841
215. Bertini L., Sole A.D., Gabrielli D., Jona-Lasinio G., and Landim C. Stochastic interacting particle systems out of equilibrium. J. Stat. Mech. 2007 07 (2007) P07014
216. Kipnis C., Olla S., and Varadhan S.R.S. Hydrodynamics and large deviation for simple exclusion processes. Comm. Pure Appl. Math. 42 2 (1989) 115-137
217. Derrida B., and Lebowitz J.L. Exact large deviation function in the asymmetric exclusion process. Phys. Rev. Lett. 80 2 (1998) 209-213
218. Derrida B. An exactly soluble non-equilibrium system: The asymmetric simple exclusion process. Phys. Rep. 301 1-3 (1998) 65-83
219. Benois O., Kipnis C., and Landim C. Large deviations from the hydrodynamical limit of mean zero asymmetric zero range processes. Stochastic Process. Appl. 55 1 (1995) 65-89
220. Landim C. Hydrodynamical limit for mean zero asymmetric zero range processes. In: Boccara N., Goles E., Martinez S., and Picco P. (Eds). Cellular Automata and Cooperative Systems (1993), Kluwer, Boston
221. Kipnis C., and Olla S. Large deviations from the hydrodynamical limit for a system of independent brownian particles. Stochastics 33 1 (1990) 17-25
222. Kipnis C., and Léonard C. Grandes déviations pour un système hydrodynamique asymétrique de particules indépendantes. Ann. Inst. Poincaré B 31 1 (1995) 223-248
223. Landim C. An overview on large deviations of the empirical measure of interacting particle systems. Ann. Inst. Poincaré A 55 2 (1991) 615-635
224. Derrida B., Lebowitz J.L., and Speer E.R. Free energy functional for nonequilibrium systems: An exactly solvable case. Phys. Rev. Lett. 87 15 (2001) 150601
225. Derrida B., Lebowitz J.L., and Speer E.R. Large deviation of the density profile in the steady state of the open symmetric simple exclusion process. J. Statist. Phys. 107 3 (2002) 599-634
226. Derrida B., Lebowitz J.L., and Speer E.R. Exact free energy functional for a driven diffusive open stationary nonequilibrium system. Phys. Rev. Lett. 89 3 (2002) 030601
227. Derrida B., Lebowitz J.L., and Speer E.R. Exact large deviation functional of a stationary open driven diffusive system: The asymmetric exclusion process. J. Statist. Phys. 110 3 (2003) 775-810
228. Bodineau T., and Derrida B. Current fluctuations in nonequilibrium diffusive systems: An additivity principle. Phys. Rev. Lett. 92 18 (2004) 180601
229. Bodineau T., and Derrida B. Distribution of current in nonequilibrium diffusive systems and phase transitions. Phys. Rev. E 72 6 (2005) 066110
230. Bodineau T., and Derrida B. Current large deviations for asymmetric exclusion processes with open boundaries. J. Statist. Phys. 123 2 (2006) 277-300
231. Bodineau T., and Giacomin G. From dynamic to static large deviations in boundary driven exclusion particle systems. Stochastic Process. Appl. 110 1 (2004) 67-81
232. Paladin G., and Vulpiani A. Anomalous scaling laws in multifractal objects. Phys. Rep. 156 (1987) 147-225
233. McCauley J.L. Introduction to multifractals in dynamical systems theory and fully developed fluid turbulence. Phys. Rep. 189 5 (1990) 225-266
234. Falconer K. Techniques in Fractal Geometry (1997), Wiley, New York
235. Zohar G. Large deviations formalism for multifractals. Stochastic Process. Appl. 79 (1999) 229-242
236. Veneziano D. Large deviations of multifractal measures. Fractals 10 (2002) 117-129
237. Harte D. Multifractals: Theory and Applications (2001), CRC Press, New York
238. Tominaga H., Hata H., Horita T., Mori H., and Tomita K. Linearities of the f (α) spectrum at bifurcations of chaos in dissipative differential systems. Progr. Theoret. Phys. 84 (1990) 18-22
239. Hata H., Horita T., Mori H., Morita T., and Tomita K. Singular local structures of chaotic attractors and q-phase transitions of spatial scaling structures. Progr. Theoret. Phys. 81 (1989) 11-16
240. Touchette H., and Beck C. Nonconcave entropies in multifractals and the thermodynamic formalism. J. Statist. Phys. 125 (2006) 455-471
241. Riedi R. An improved multifractal formalism and self-similar measures;. J. Math. Anal. Appl. 189 2 (1995) 462-490
242. Testud B. Transitions de phase dans l'analyse multifractale de mesures auto-similaires. C.R. Acad. Sci. Paris Ser. I 340 9 (2005) 653-658
243. Testud B. Phase transitions for the multifractal analysis of self-similar measures. Nonlinearity 19 5 (2006) 1201-1217
244. Kifer Y. Random perturbations of dynamical systems. Progress in Probability and Statistics vol. 16 (1988), Birkhäuser, Boston
245. Kifer Y. Large deviations in dynamical systems and stochastic processes. Trans. Amer. Math. Soc. 321 2 (1990) 505-524
246. Alekseev V.M., and Yakobson M.V. Symbolic dynamics and hyperbolic dynamic systems. Phys. Rep. 75 (1981) 290-325
247. Eckmann J.-P., and Ruelle D. Ergodic theory of chaos and strange attractors. Rev. Modern Phys. 57 (1985) 617-656
248. Lasota A., and Mackey M.C. Chaos, Fractals, and Noise: Stochastic Aspects of Dynamics. Applied Mathematical Sciences vol. 97 (1994), Springer, New York
249. Ruelle D. The thermodynamic formalism for expanding maps. Comm. Math. Phys. 125 2 (1989) 239-262
250. Sinai Y.G. Gibbs measures in ergodic theory. Russ. Math. Surveys 27 4 (1972) 21-69
251. Sinai Y.G. Topics in Ergodic Theory (1994), Princeton University Press, Princeton
252. Keller G. Equilibrium States in Ergodic Theory. London Math. Soc. Student Texts vol. 42 (1998), Cambridge University Press, Cambridge
253. Kai T., and Tomita K. Statistical mechanics of deterministic chaos: The case of one-dimensional discrete process. Progr. Theoret. Phys. 64 (1980) 1532-1550
254. Mori H., Hata H., Horita T., and Kobayashi T. Statistical mechanics of dynamical systems. Progr. Theoret. Phys. Suppl. 99 (1989) 1-63
255. Takahashi Y., and Oono Y. Towards the statistical mechanics of chaos. Progr. Theoret. Phys. 71 (1984) 851-854
256. Young L.-S. Some large deviation results for dynamical systems. Trans. Amer. Math. Soc. 318 2 (1990) 525-543
257. Lopes A.O. Entropy and large deviation. Nonlinearity 3 (1990) 527-546
258. Waddington S. Large deviation asymptotics for Anosov flows. Ann. Inst. Poincaré C 13 4 (1996) 445-484
259. Pollicott M., Sharp R., and Yuri M. Large deviations for maps with indifferent fixed points. Nonlinearity 11 (1998) 1173-1184
260. Young L.-S. Entropy in dynamical systems. In: Greven A., Keller G., and Warnecke G. (Eds). Entropy. Princeton Ser. Appl. Math. (2003), Princeton University Press, Princeton 313-327 (Chapter 16)
261. Maes C., and Verbitskiy E. Large deviations and a fluctuation symmetry for chaotic homeomorphisms. Comm. Math. Phys. 233 (2003) 137-151
262. J. Tailleur, Grandes déviations, physique statistique et systèmes dynamiques, Ph.D. Thesis, Université Pierre et Marie Curie, Paris, 2007
263. Oono Y. A heuristic approach to the Kolmogorov entropy as a disorder parameter. Progr. Theoret. Phys. 60 (1978) 1944-1946
264. Oono Y., and Takahashi Y. Chaos, external noise and Fredholm theory. Progr. Theoret. Phys. 63 (1980) 1804-1807
265. Gantert N., and Zeitouni O. Large deviations for one-dimentional random walk in a random environment - A survey. In: Revesz P., and Toth B. (Eds). Proceedings of the Conference on Random Walks. Bolyai Society Mathematical Studies vol. 9 (1999) 127-165
266. Comets F., Gantert N., and Zeitouni O. Quenched, annealed and functional large deviations for one-dimensional random walk in random environment. Probab. Theory Related Fields 118 1 (2000) 65-114
267. Varadhan S.R.S. Large deviations for random walks in a random environment. Comm. Pure Appl. Math. 56 8 (2003) 1222-1245
268. Zeitouni O. Random walks in random environments. J. Phys. A: Math. Gen. 39 40 (2006) R433-R464
269. Dorlas T.C., and Wedagedera J.R. Large deviations and the random energy model. Internat. J. Modern Phys. B 15 (2001) 1-15
270. Dorlas T.C., and Dukes W.M.B. Large deviation approach to the generalized random energy model. J. Phys. A: Math. Gen. 35 20 (2002) 4385-4394
271. Talagrand M. Large deviations, Guerra's and A.S.S. schemes, and the Parisi hypothesis. J. Statist. Phys. 126 4 (2007) 837-894
272. Talagrand M. Spin Glasses: A Challenge for Mathematicians (2003), Springer, New York
273. Bovier A. Statistical Mechanics of Disordered Systems: A Mathematical Perspective (2006), Cambridge University Press, Cambridge
274. Mézard M., and Montanari A. Information, Physics, and Computation (2009), Oxford University Press
275. Cegla W., Lewis J.T., and Raggio G.A. The free energy of quantum spin systems and large deviations. Comm. Math. Phys. 118 2 (1988) 337-354
276. van den Berg M., Lewis J.T., and Pule J.V. The large deviation principle and some models of an interacting boson gas. Comm. Math. Phys. 118 1 (1988) 61-85
277. Dorlas T.C., Martin P., and Pule J. Long cycles in a perturbed mean field model of a boson gas. J. Statist. Phys. 121 3 (2005) 433-461
278. M. Lenci, Classical billiards and quantum large deviations, Ph.D. Thesis, Rutgers University, New Brunswick, N.J., 1999
279. Lebowitz J.L., Lenci M., and Spohn H. Large deviations for ideal quantum systems. J. Math. Phys. 41 3 (2000) 1224-1243
280. Gallavotti G., Lebowitz J.L., and Mastropietro V. Large deviations in rarefied quantum gases. J. Statist. Phys. 108 5 (2002) 831-861
281. Hiai F., Mosonyi M., and Ogawa T. Large deviations and Chernoff bound for certain correlated states on a spin chain. J. Math. Phys. 48 12 (2007) 123301
282. Lenci M., and Rey-Bellet L. Large deviations in quantum lattice systems: One-phase region. J. Statist. Phys. 119 3 (2005) 715-746
283. Netočný K., and Redig F. Large deviations for quantum spin systems. J. Statist. Phys. 117 3 (2004) 521-547
284. Petz D., Raggio G.A., and Verbeure A. Asymptotics of Varadhan-type and the Gibbs variational principle. Comm. Math. Phys. 121 2 (1989) 271-282
285. Bjelaković I., Deuschel J.-D., Krüger T., Seiler R., Siegmund-Schultze R., and Szkola A. A quantum version of Sanov's Theorem. Comm. Math. Phys. 260 3 (2005) 659-671
286. Keyl M. Quantum state estimation and large deviations. Rev. Math. Phys. 18 1 (2006) 19-60
287. Audenaert K.M.R., Calsamiglia J., Muñoz-Tapia R., Bagan E., Masanes L., Acin A., and Verstraete F. Discriminating states: The quantum chernoff bound. Phys. Rev. Lett. 98 16 (2007) 160501
288. Ahlswede R., and Blinovsky V.M. Large deviations in quantum information theory. Probab. Inform. Trans. 39 4 (2003) 373-379
289. Blanchard P., Combe P., Sirugue M., and Sirugue-Collin M. Estimates of quantum deviations from classical mechanics using large deviation results. Quantum Probability and Applications II. Lecture Notes in Mathematics vol. 1136 (1985), Springer 104-111
290. Ellis R.S. The theory of large deviations and applications to statistical mechanics. Lectures for the International Seminar on Extreme Events in Complex Dynamics (2006)
291. Fowler R.H. Statistical Mechanics. 2nd ed. (1966), Cambridge University Press, Cambridge
292. Feller W. An Introduction to Probability Theory and its Applications, vol. II (1970), Wiley, New York