The European Physical Journal B

, Volume 61, Issue 1, pp 1–24 | Cite as

Heat and fluctuations from order to chaos

  • G. Gallavotti


The Heat theorem reveals the second law of equilibrium Thermodynamics (i.e. existence of Entropy) as a manifestation of a general property of Hamiltonian Mechanics and of the Ergodic Hypothesis, valid for 1 as well as 1023 degrees of freedom systems, i.e. for simple as well as very complex systems, and reflecting the Hamiltonian nature of the microscopic motion. In Nonequilibrium Thermodynamics theorems of comparable generality do not seem to be available. Yet it is possible to find general, model independent, properties valid even for simple chaotic systems (i.e. the hyperbolic ones), which acquire special interest for large systems: the Chaotic Hypothesis leads to the Fluctuation Theorem which provides general properties of certain very large fluctuations and reflects the time-reversal symmetry. Implications on Fluids and Quantum systems are briefly hinted. The physical meaning of the Chaotic Hypothesis, of SRB distributions and of the Fluctuation Theorem is discussed in the context of their interpretation and relevance in terms of Coarse Grained Partitions of phase space. This review is written taking some care that each section and appendix is readable either independently of the rest or with only few cross references.


05.20.-y Classical statistical mechanics 05.70.Ln Nonequilibrium and irreversible thermodynamics 05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. S.G. Brush, History of modern physical sciences: The kinetic theory of gases (Imperial College Press, London, 2003) Google Scholar
  2. M.W. Zemansky, Heat and thermodynamics (McGraw-Hill, New-York, 1957) Google Scholar
  3. R.P. Feynman, R.B. Leighton, M. Sands, The Feynman lectures in Physics (Addison-Wesley, New York, 1963), Vol. I, II, III Google Scholar
  4. L. Boltzmann, Über die mechanische Bedeutung des zweiten Haupsatzes der Wärmetheorie, Volume 1, p. 9 of Wissenschaftliche Abhandlungen, edited by F. Hasenöhrl (Chelsea, New York, 1968) Google Scholar
  5. L. Boltzmann, Über die Eigenshaften monozyklischer und anderer damit verwandter Systeme, Volume 3, p. 122 of Wissenschaftliche Abhandlungen (Chelsea, New-York, 1968) Google Scholar
  6. G.N. Bochkov, Yu.E. Kuzovlev, Physica A 106, 443 (1981) CrossRefADSMathSciNetGoogle Scholar
  7. C. Jarzynski, Phys. Rev. Lett. 78, 2690 (1997) CrossRefADSGoogle Scholar
  8. C. Jarzynski, J. Stat. Phys. 98, 77 (1999) CrossRefMathSciNetGoogle Scholar
  9. D.J. Evans, D.J. Searles, Phys. Rev. E 50, 1645 (1994). CrossRefADSGoogle Scholar
  10. G.E. Crooks, Phys. Rev. E 60, 2721 (1999) CrossRefADSGoogle Scholar
  11. E.G.D. Cohen, G. Gallavotti, J. Stat. Phys. 96, 1343 (1999) zbMATHCrossRefGoogle Scholar
  12. G. Gallavotti, Entropy, nonequilibrium, chaos and infinitesimals, e-print arXiv:cond-mat/0606477 (2006) Google Scholar
  13. R.P. Feynman, F.L. Vernon, Ann. Phys. 24, 118 (1963) CrossRefADSMathSciNetGoogle Scholar
  14. K. Hepp, E. Lieb, Phys. Rev. A 8, 2517 (1973) CrossRefADSMathSciNetGoogle Scholar
  15. J.P. Eckmann, C.A. Pillet, L.R. Bellet, Comm. Math. Phys. 201, 657 (1999) zbMATHCrossRefADSMathSciNetGoogle Scholar
  16. W. Aschbacher, Y. Pautrat, V. Jakšić, C.A. Pillet, Lect. Notes Math. 1882, 1 (2006) MathSciNetCrossRefGoogle Scholar
  17. P. Hänggi, G.L. Ingold, Chaos 15, 026105 (2005) CrossRefMathSciNetADSGoogle Scholar
  18. D. Abraham, E. Baruch, G. Gallavotti, A. Martin-Löf, Studies Appl. Math. 51, 211 (1972) Google Scholar
  19. E. Barouch, M. Dresden, Phys. Rev. Lett. 23, 114 (1969) CrossRefADSGoogle Scholar
  20. J.L. Lebowitz, Physics Today 32 (1993) Google Scholar
  21. G. Gallavotti, Comm. Math. Phys. 224, 107 (2001) zbMATHCrossRefADSMathSciNetGoogle Scholar
  22. D. Ruelle, Comm. Math. Phys. 189, 365 (1997) zbMATHCrossRefADSMathSciNetGoogle Scholar
  23. G. Gallavotti, Statistical Mechanics. A short treatise (Springer Verlag, Berlin, 2000) Google Scholar
  24. D. Ruelle, Progress in Theoretical Physics Supplement 64, 339 (1978) CrossRefADSMathSciNetGoogle Scholar
  25. D. Ruelle, Ergodic theory, Volume Suppl X of The Boltzmann equation, edited by E.G.D Cohen, W. Thirring, Acta Physica Austriaca (Springer, New York, 1973) Google Scholar
  26. G. Gallavotti, E.G.D. Cohen, Phys. Rev. Lett. 74, 2694 (1995) CrossRefADSGoogle Scholar
  27. D.J. Evans, E.G.D. Cohen, G.P. Morriss, Phys. Rev. Lett. 71, 2401 (1993) zbMATHCrossRefADSGoogle Scholar
  28. G. Gallavotti, F. Bonetto, G. Gentile, Aspects of the ergodic, qualitative and statistical theory of motion (Springer Verlag, Berlin, 2004) Google Scholar
  29. G. Gallavotti, J. Stat. Phys. 78, 1571 (1995) zbMATHCrossRefADSMathSciNetGoogle Scholar
  30. Ya.G. Sinai, Russian Math. Surveys 27, 21 (1972) zbMATHCrossRefADSMathSciNetGoogle Scholar
  31. R. Bowen, D. Ruelle, Invent. Math. 29, 181 (1975) zbMATHCrossRefADSMathSciNetGoogle Scholar
  32. Ya.G. Sinai, Lectures in ergodic theory, Lecture notes in Mathematics (Princeton University Press, Princeton, 1977) Google Scholar
  33. W. Thomson, in Proceedings of the Royal Society of Edinburgh 8, 325 (1874) Google Scholar
  34. L. Boltzmann, Einige allgemeine sätze über Wärme-gleichgewicht, Volume 1, p. 259 of Wissenschaftliche Abhandlungen, edited by F. Hasenöhrl (Chelsea, New York, 1968) Google Scholar
  35. L. Boltzmann, Studien über das Gleichgewicht der le-bendigen Kraft zwischen bewegten materiellen Punkten, Volume 1, p. 49 of Wissenschaftliche Abhandlungen, edited by F. Hasenöhrl (Chelsea, New York, 1968) Google Scholar
  36. H. Helmholtz, Prinzipien der Statistik monocyklischer Systeme, Volume III of Wissenschaftliche Abhandlungen (Barth, Leipzig, 1895) Google Scholar
  37. H. Helmholtz, Studien zur Statistik monocyklischer Systeme, Volume III of Wissenschaftliche Abhandlungen (Barth, Leipzig, 1895) Google Scholar
  38. L. Boltzmann, Reply to Zermelo's Remarks on the theory of heat, Volume 1, p. 392 of History of modern physical sciences: The kinetic theory of gases, edited by S. Brush (Imperial College Press, London, 2003) Google Scholar
  39. G. Gallavotti, Math. Phys. Electronic J. (MPEJ) 1, 1 (1995) MathSciNetGoogle Scholar
  40. G. Gallavotti, E.G.D. Cohen, J. Stat. Phys. 80, 931 (1995) zbMATHCrossRefADSMathSciNetGoogle Scholar
  41. G. Gallavotti, E.G.D. Cohen, Phys. Rev. E 69, 035104 (2004) CrossRefADSMathSciNetGoogle Scholar
  42. A. Giuliani, F. Zamponi, G. Gallavotti, J. Stat. Phys. 119, 909 (2005) zbMATHCrossRefMathSciNetADSGoogle Scholar
  43. G. Gentile, Forum Math. 10, 89 (1998) zbMATHMathSciNetGoogle Scholar
  44. G. Gallavotti, Phys. Rev. Lett. 77, 4334 (1996) zbMATHCrossRefADSMathSciNetGoogle Scholar
  45. D. Ruelle, J. Stat. Phys. 95, 393 (1999) zbMATHCrossRefMathSciNetGoogle Scholar
  46. G. Gallavotti, Annales de l'Institut H. Poincaré 70, 429 (1999) and chao-dyn/9703007 zbMATHMathSciNetGoogle Scholar
  47. G. Gallavotti, Open Systems and Information Dynamics 6, 101 (1999) zbMATHCrossRefMathSciNetGoogle Scholar
  48. R. Chetrite, J.Y. Delannoy, K. Gawedzki, J. Stat. Phys. 126, 1165 (2007) zbMATHCrossRefADSMathSciNetGoogle Scholar
  49. N.I. Chernov, G.L. Eyink, J.L. Lebowitz, Ya.G. Sinai, Comm. Math. Phys. 154, 569 (1993) zbMATHCrossRefADSMathSciNetGoogle Scholar
  50. F. Bonetto, G. Gallavotti, P. Garrido, Physica D 105, 226 (1997) zbMATHCrossRefADSMathSciNetGoogle Scholar
  51. G. Gallavotti, Chaos 16, 043114 (2006) CrossRefMathSciNetADSGoogle Scholar
  52. R. Becker, Electromagnetic fields and interactions (Blaisdell, New-York, 1964) Google Scholar
  53. G. Gallavotti, Foundations of Fluid Dynamics (second printing) (Springer Verlag, Berlin, 2005) Google Scholar
  54. G. Gallavotti, Physica D 105, 163 (1997) zbMATHCrossRefADSMathSciNetGoogle Scholar
  55. L. Rondoni, E. Segre, Nonlinearity 12, 1471 (1999) zbMATHCrossRefADSMathSciNetGoogle Scholar
  56. S. de Groot, P. Mazur, Non equilibrium thermodynamics (Dover, Mineola, NY, 1984) Google Scholar
  57. G. Gallavotti, Chaos 16, 023130 (2006) ADSGoogle Scholar
  58. G. Gallavotti, Quantum nonequilibrium and entropy creation, e-print arXiv:cond-mat/0701124, Unpublished, 2007 Google Scholar
  59. D.J. Evans, G.P. Morriss, Statistical Mechanics of Nonequilibrium Fluids (Academic Press, New-York, 1990) Google Scholar
  60. L.F. Cugliandolo, J. Kurchan, L. Peliti, Phys. Rev. E 2898 (1997) Google Scholar
  61. A. Crisanti, F. Ritort, J. Phys. A R181 (2003) Google Scholar
  62. S. Lepri, R. Livi, A. Politi, Physica D 119, 140 (1998) CrossRefADSMathSciNetGoogle Scholar
  63. R. Van Zon, E.G.D. Cohen, Phys. Rev. Lett. 91, 110601 (2003) CrossRefGoogle Scholar
  64. F. Bonetto, G. Gallavotti, A. Giuliani, F. Zamponi, J. Stat. Phys. 123, 39 (2006) zbMATHCrossRefMathSciNetADSGoogle Scholar
  65. F. Bonetto, G. Gallavotti, G. Gentile, Ergodic Theory and Dynamical Systems, 28, 21–47 (2008); doi: 10.1017/S0143385707000417 zbMATHMathSciNetGoogle Scholar
  66. M. Bandi, J.R. Cressman, W. Goldburg, J. Stat. Phys. 130, 27–38 (2008); doi: 10.1007/s10955-007-9355-4 CrossRefADSzbMATHMathSciNetGoogle Scholar
  67. F. Bonetto, G. Gallavotti, A. Giuliani, F. Zamponi, J. Stat. Mech. P05009 (2006) Google Scholar
  68. J. Kurchan, J. Phys. A 31, 3719 (1998). zbMATHCrossRefADSMathSciNetGoogle Scholar
  69. J. Lebowitz, H. Spohn, J. Stat. Phys. 95, 333 (1999) zbMATHCrossRefMathSciNetGoogle Scholar
  70. C. Maes, J. Stat. Phys. 95, 367 (1999) zbMATHCrossRefMathSciNetGoogle Scholar
  71. J. Kurchan, A quantum fluctuation theorem, e-print arXiv:cond-mat/0007360, unpublished (2000) Google Scholar
  72. V. Jakšić, C.A. Pillet, Comm. Math. Phys. 226, 131 (2002) CrossRefADSMathSciNetzbMATHGoogle Scholar
  73. V. Jakšić, Y. Ogata, C.A. Pillet, Ann. Henri Poincaré (2007) Google Scholar
  74. J.R. Cressman, J. Davoudi, W.I. Goldburg, J. Schumacher, New J. Phys. 6, 53 (2004) CrossRefADSMathSciNetGoogle Scholar
  75. F. Bonetto, G. Gallavotti, Comm. Math. Phys. 189, 263 (1997) zbMATHCrossRefADSMathSciNetGoogle Scholar
  76. F. Zamponi, Is it possible to experimentally verify the fluctuation relation? A review of theoretical motivations and numerical evidence, e-print arXiv:cond-mat/0612019 (2006) Google Scholar
  77. G. Gallavotti, Chaos 14, 680 (2004) zbMATHCrossRefADSMathSciNetGoogle Scholar
  78. R. Bowen, in Proceedings of the American Mathematical Society 71, 130 (1978) zbMATHCrossRefMathSciNetGoogle Scholar
  79. D. Levesque, L. Verlet, J. Stat. Phys. 72, 519 (1993) zbMATHCrossRefADSGoogle Scholar
  80. R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeormorphisms, Volume 470 of Lecture Notes in Mathematics (Springer-Verlag, Berlin-Heidelberg, 1975) Google Scholar
  81. E. Lorenz, J. Atmospheric Sci. 20, 130 (1963) CrossRefADSGoogle Scholar
  82. P. Garrido, G. Gallavotti, J. Stat. Phys. 126, 1201 (2007) zbMATHCrossRefADSMathSciNetGoogle Scholar
  83. D. Dürr, S. Goldstein, N. Zanghì. J. Stat. Phys. 68, 259 (1992) zbMATHCrossRefADSGoogle Scholar

Copyright information

© EDP Sciences/Società Italiana di Fisica/Springer-Verlag 2008

Authors and Affiliations

  1. 1.Dipartimento di Fisica and INFNUniversità di RomaRomaItaly

Personalised recommendations