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Revisiting classical and quantum disordered systems from the unifying perspective of large deviations

  • Cécile MonthusEmail author
Colloquium
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Part of the following topical collections:
  1. Topical issue: Recent Advances in the Theory of Disordered Systems

Abstract

The theory of large deviations is already the natural language for the statistical physics of equilibrium and non-equilibrium. In the field of disordered systems, the analysis via large deviations is even more useful to describe within a unified perspective the typical events and the rare events that occur on various scales. In the present pedagogical introduction, we revisit various emblematic classical and quantum disordered systems in order to highlight the common underlying mechanisms from the point of view of large deviations.

Graphical abstract

References

  1. 1.
    Y. Oono, Prog. Theor. Phys. Suppl. 99, 165 (1989) ADSCrossRefGoogle Scholar
  2. 2.
    R.S. Ellis, Physica D 133, 106 (1999) ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    H. Touchette, Phys. Rep. 478, 1 (2009) ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    B. Derrida, J. Stat. Mech. 2007, P07023 (2007) CrossRefGoogle Scholar
  5. 5.
    R.J. Harris, G.M. Schütz, J. Stat. Mech. 2007, P07020 (2007) CrossRefGoogle Scholar
  6. 6.
    E.M. Sevick, R. Prabhakar, S.R. Williams, D.J. Searles, Annu. Rev. Phys. Chem. 59, 603 (2008) ADSCrossRefGoogle Scholar
  7. 7.
    H. Touchette, R.J. Harris, Large deviation approach to nonequilibrium systems, in Nonequilibrium Statistical Physics of Small Systems: Fluctuation Relations and Beyond (Wiley-VCH Verlag GmbH, Weinheim, 2013) Google Scholar
  8. 8.
    L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio, C. Landim, Rev. Mod. Phys. 87, 593 (2015) ADSCrossRefGoogle Scholar
  9. 9.
    A. Lazarescu, J. Phys. A: Math. Theor. 48, 503001 (2015) MathSciNetCrossRefGoogle Scholar
  10. 10.
    A. Lazarescu, J. Phys. A: Math. Theor. 50, 254004 (2017) ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    A. de La Fortelle, Contributions to the theory of large deviations and applications, Ph.D. thesis, INRIA Rocquencourt, France, 2000 Google Scholar
  12. 12.
    V. Lecomte, Thermodynamique des histoires et fluctuations hors d’équilibre, Ph.D. thesis, Université Paris 7, France, 2007 Google Scholar
  13. 13.
    R. Chétrite, Grandes déviations et relations de fluctuation dans certains modèles de systèmes hors d’équilibre, Ph.D. thesis, École Normale Supérieure de Lyon, France, 2008 Google Scholar
  14. 14.
    B. Wynants, Structures of Nonequilibrium Fluctuations, Ph.D. thesis, Catholic University of Leuven, Belgium, 2010, arXiv:1011.4210
  15. 15.
    R. Chétrite, Pérégrinations sur les phénomènes aléatoires dans la nature, HDR thesis, Université de Nice-Sophia-Antipolis, France, 2018 Google Scholar
  16. 16.
    J.M. Luck, A. Saclay, Systèmes désordonnés unidimensionnels (Commissariat à l’Atomique, France, 1992) Google Scholar
  17. 17.
    A. Crisanti, G. Paladin, A. Vulpiani, Products of Random Matrices in Statistical Physics (Springer Verlag, Berlin, 1993) Google Scholar
  18. 18.
    I.M. Lifshitz, Adv. Phys. 13, 483 (1964) ADSCrossRefGoogle Scholar
  19. 19.
    I.M. Lifshitz, Sov. Phys. Usp. 7, 549 (1965) ADSCrossRefGoogle Scholar
  20. 20.
    I.M. Lifshitz, S.A. Gredeskul, L.A. Pastur, Introduction to the Theory of Disordered Systems (John Wiley and Sons, Chichester, 1987) Google Scholar
  21. 21.
    B.I. Shklovskii, A.L. Efros, Electronic Properties of Doped Semiconductors (Springer Verlag, Berlin, 1984) Google Scholar
  22. 22.
    R.B. Griffiths, Phys. Rev. Lett. 23, 17 (1969) ADSCrossRefGoogle Scholar
  23. 23.
    Y. Kafri, D. Mukamel, Phys. Rev. Lett. 91, 055502 (2003) ADSCrossRefGoogle Scholar
  24. 24.
    M. Randeria, J.P. Sethna, R.G. Palmer, Phys. Rev. Lett. 54, 1321 (1985) ADSCrossRefGoogle Scholar
  25. 25.
    A.J. Bray, Phys. Rev. Lett. 59, 586 (1987) ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    A.J. Bray, Phys. Rev. Lett. 60, 720 (1988) ADSMathSciNetCrossRefGoogle Scholar
  27. 27.
    F. Iglói, C. Monthus, Phys. Rep. 412, 277 (2005) ADSCrossRefGoogle Scholar
  28. 28.
    F. Iglói, C. Monthus, Eur. Phys. J. B 91, 290 (2018) ADSCrossRefGoogle Scholar
  29. 29.
    M. Janssen, Phys. Rep. 295, 1 (1998) ADSCrossRefGoogle Scholar
  30. 30.
    F. Evers, A.D. Mirlin, Rev. Mod. Phys. 80, 1355 (2008) ADSCrossRefGoogle Scholar
  31. 31.
    A.W.W. Ludwig, Nucl. Phys. B 330, 639 (1990) ADSCrossRefGoogle Scholar
  32. 32.
    J.L. Jacobsen, J.L. Cardy, Nucl. Phys. B 515, 701 (1998) ADSCrossRefGoogle Scholar
  33. 33.
    C. Chatelain, B. Berche, Nucl. Phys. B 572, 626 (2000) ADSCrossRefGoogle Scholar
  34. 34.
    C. Chatelain, B. Berche, L.N. Shchur, J. Phys. A: Math. Gen. 34, 9593 (2001) ADSCrossRefGoogle Scholar
  35. 35.
    G. Palágyi, C. Chatelain, B. Berche, F. Iglói, Eur. Phys. J. B 13, 357 (2000) ADSCrossRefGoogle Scholar
  36. 36.
    B. Berche, C. Chatelain, in Order, Disorder, and Criticality, edited by Yu. Holovatch (World Scientific, Singapore, 2004), p. 146 Google Scholar
  37. 37.
    M.J. Thill, H.J. Hilhorst, J. Phys. I 6, 67 (1996) Google Scholar
  38. 38.
    C. Monthus, B. Berche, C. Chatelain, J. Stat. Mech. 2009, 12002 (2009) CrossRefGoogle Scholar
  39. 39.
    R.S. Ellis, Entropy, Large Deviations, and Statistical Mechanics (Springer Verlag, New York, 1985) Google Scholar
  40. 40.
    J.D. Deuschel, D.W. Stroock, Large Deviations (Academic Press, Boston, 1989) Google Scholar
  41. 41.
    A. Dembo, O. Zeitouni, Large Deviations Techniques and Applications (Springer Verlag, Berlin, 1998) Google Scholar
  42. 42.
    F. den Hollander, Large Deviations, Fields Institute Monographs (American Mathematical Society, Providence, 2008) Google Scholar
  43. 43.
    R. Azencott, M.I. Freidlin, S.R.S. Varadhan, Large Deviations at Saint-Flour, Probability at Saint-Flour (Springer, Heidelberg, 2012) Google Scholar
  44. 44.
    F. Rassoul-Agha, T. Seppalainen, A Course on Large Deviations with an Introduction to Gibbs Measures, Graduate Studies in Mathematics (American Mathematical Society, Providence, 2015), Vol. 162 Google Scholar
  45. 45.
    M. Talagrand, Spin-Glasses: A Challenge for Mathematicians (Springer, New York, 2003) Google Scholar
  46. 46.
    A. Bovier, Statistical Mechanics of Disordered Systems: A Mathematical Perspective (Cambridge University Press, Cambridge, 2006) Google Scholar
  47. 47.
    F. Comets, Directed Polymers in Random Environments, Probability in Saint-Flour, Lecture Notes in Mathematics 2175 (Springer International Publishing, Cham, 2017) Google Scholar
  48. 48.
    O. Zeitouni, J. Phys. A: Math. Gen. 39, R433 (2006) ADSCrossRefGoogle Scholar
  49. 49.
    A. Comtet, Y. Tourigny, arXiv:1601.01822
  50. 50.
    B. Derrida, H. Hilhorst, J. Phys. C: Solid State Phys. 14, L539 (1981) ADSCrossRefGoogle Scholar
  51. 51.
    C. Monthus, J. Phys. A: Math. Theor. 51, 465301 (2018) ADSMathSciNetCrossRefGoogle Scholar
  52. 52.
    B.L. Altshuler, Y. Gefen, A. Kamenev, L.S. Levitov, Phys. Rev. Lett. 78, 2803 (1997) ADSCrossRefGoogle Scholar
  53. 53.
    A. De Luca, B.L. Altshuler, V.E. Kravtsov, A. Scardicchio, Phys. Rev. Lett. 113, 046806 (2014) ADSCrossRefGoogle Scholar
  54. 54.
    F. Pietracaprina, V. Ros, A. Scardicchio, Phys. Rev. B 93, 054201 (2016) ADSCrossRefGoogle Scholar
  55. 55.
    C. Monthus, J. Stat. Mech. 2016, 123303 (2016) CrossRefGoogle Scholar
  56. 56.
    L.B. Ioffe, M. Mézard, Phys. Rev. Lett. 105, 037001 (2010) ADSCrossRefGoogle Scholar
  57. 57.
    M.V. Feigelman, L.B. Ioffe, M. Mézard, Phys. Rev. B 82, 184534 (2010) ADSCrossRefGoogle Scholar
  58. 58.
    O. Dimitrova, M. Mézard, J. Stat. Mech. 2011, P01020 (2011) CrossRefGoogle Scholar
  59. 59.
    G. Fayolle, A. de La Fortelle, Problems of Information Transmission 38, 354 (2002) MathSciNetCrossRefGoogle Scholar
  60. 60.
    A. de La Fortelle, Problems of Information Transmission 37, 120 (2001) MathSciNetCrossRefGoogle Scholar
  61. 61.
    C. Maes, K. Netocny, Europhys. Lett. 82, 30003 (2008) ADSCrossRefGoogle Scholar
  62. 62.
    C. Maes, K. Netocny, B. Wynants, Markov Proc. Rel. Fields. 14, 445 (2008) Google Scholar
  63. 63.
    A.C. Barato, R. Chetrite, J. Stat. Phys. 160, 1154 (2015) ADSMathSciNetCrossRefGoogle Scholar
  64. 64.
    L. Bertini, A. Faggionato, D. Gabrielli, Ann. Inst. Henri Poincaré Probab. Statist. 51, 867 (2015) ADSCrossRefGoogle Scholar
  65. 65.
    L. Bertini, A. Faggionato, D. Gabrielli, Stoch. Process. Appl. 125, 2786 (2015) CrossRefGoogle Scholar
  66. 66.
    C. Monthus, J. Stat. Mech. 2019, 023206 (2019) CrossRefGoogle Scholar
  67. 67.
    C. Monthus, J. Phys. A: Math. Theor. 52, 025001 (2019) ADSMathSciNetCrossRefGoogle Scholar
  68. 68.
    C. Monthus, J. Phys. A: Math. Theor. 52, 135003 (2019) ADSMathSciNetCrossRefGoogle Scholar
  69. 69.
    C. Maes, K. Netocny, B. Wynants, Physica A 387, 2675 (2008) ADSMathSciNetCrossRefGoogle Scholar
  70. 70.
    J. Hoppenau, D. Nickelsen, A. Engel, New J. Phys. 18, 083010 (2016) ADSCrossRefGoogle Scholar
  71. 71.
    B. Derrida, H. Spohn, J. Stat. Phys. 51, 817 (1988) ADSCrossRefGoogle Scholar
  72. 72.
    J. Cook, B. Derrida, J. Stat. Phys. 63, 505 (1991) ADSCrossRefGoogle Scholar
  73. 73.
    R. Abou-Chacra, P.W. Anderson, D.J. Thouless, J. Phys. C: Solid State Phys. 6, 1734 (1973) ADSCrossRefGoogle Scholar
  74. 74.
    R. Abou-Chacra, D.J. Thouless, J. Phys. C: Solid State Phys. 7, 65 (1974) ADSCrossRefGoogle Scholar
  75. 75.
    B. Derrida, G.J. Rodgers, J. Phys. A: Math. Gen. 26, L457 (1993) ADSCrossRefGoogle Scholar
  76. 76.
    J.D. Miller, B. Derrida, J. Stat. Phys. 75, 357 (1994) ADSCrossRefGoogle Scholar
  77. 77.
    B. Derrida, Phys. Rev. B 24, 2613 (1981) ADSMathSciNetCrossRefGoogle Scholar
  78. 78.
    E.J. Gumbel, Statistics of Extreme (Columbia University Press, New York, 1958) Google Scholar
  79. 79.
    J. Galambos, The Asymptotic Theory of Extreme Order Statistics (Krieger, Malabar, 1987) Google Scholar
  80. 80.
    B. Derrida, G. Toulouse, J. Phys. Lett. 46, L223 (1985) CrossRefGoogle Scholar
  81. 81.
    B. Derrida, H. Flyvbjerg, J. Phys. A: Math. Gen. 20, 5273 (1987) ADSCrossRefGoogle Scholar
  82. 82.
    B. Derrida, Non-self-averaging effects in sums of random variables, spin glasses, random maps and random walks, in On Three Levels, edited by M. Fannes et al. (Plenum Press, New York, 1994) Google Scholar
  83. 83.
    L.S. Levitov, Europhys. Lett. 9, 83 (1989) ADSCrossRefGoogle Scholar
  84. 84.
    L.S. Levitov, Phys. Rev. Lett. 64, 547 (1990) ADSMathSciNetCrossRefGoogle Scholar
  85. 85.
    B.L. Altshuler, L.S. Levitov, Phys. Rep. 288, 487 (1997) ADSCrossRefGoogle Scholar
  86. 86.
    L.S. Levitov, Ann. Phys. 8, 507 (1999) CrossRefGoogle Scholar
  87. 87.
    F. Evers, A.D. Mirlin, Phys. Rev. Lett. 84, 3690 (2000) ADSCrossRefGoogle Scholar
  88. 88.
    A.D. Mirlin, F. Evers, Phys. Rev. B 62, 7920 (2000) ADSCrossRefGoogle Scholar
  89. 89.
    Y.V. Fyodorov, A. Ossipov, A. Rodriguez, J. Stat. Mech. 2009, L12001 (2009) CrossRefGoogle Scholar
  90. 90.
    Y.V. Fyodorov, A. Kupiainen, C. Webb, arXiv:1509.01366
  91. 91.
    O. Yevtushenko, V.E. Kratsov, J. Phys. A 36, 8265 (2003) ADSMathSciNetCrossRefGoogle Scholar
  92. 92.
    O. Yevtushenko, A. Ossipov, J. Phys. A 40, 4691 (2007) ADSMathSciNetCrossRefGoogle Scholar
  93. 93.
    S. Kronmüller, O.M. Yevtushenko, E. Cuevas, J. Phys. A 43, 075001 (2010) ADSMathSciNetCrossRefGoogle Scholar
  94. 94.
    V.E. Kratsov, A. Ossipov, O.M. Yevtushenko, E. Cuevas, Phys. Rev. B 82, 161102(R) (2010) ADSCrossRefGoogle Scholar
  95. 95.
    E. Bogomolny, O. Giraud, Phys. Rev. E 84, 036212 (2012) ADSCrossRefGoogle Scholar
  96. 96.
    E. Bogomolny, O. Giraud, Phys. Rev. E 84, 046208 (2012) ADSCrossRefGoogle Scholar
  97. 97.
    E. Bogomolny, O. Giraud, Phys. Rev. Lett. 106, 044101 (2011) ADSCrossRefGoogle Scholar
  98. 98.
    C. Monthus, T. Garel, J. Stat. Mech. 2010, P09015 (2010) Google Scholar
  99. 99.
    C. Monthus, Entropy 18, 122 (2016) ADSCrossRefGoogle Scholar
  100. 100.
    C. Monthus, J. Stat. Mech. 2016, 073301 (2016) CrossRefGoogle Scholar
  101. 101.
    C. Monthus, J. Stat. Mech. 2017, 123304 (2017) CrossRefGoogle Scholar
  102. 102.
    B. Derrida, J.L. Lebowitz, Phys. Rev. Lett. 80, 209 (1998) ADSMathSciNetCrossRefGoogle Scholar
  103. 103.
    D.S. Dean, S.N. Majumdar, Phys. Rev. Lett. 97, 160201 (2006) ADSMathSciNetCrossRefGoogle Scholar
  104. 104.
    D.S. Dean, S.N. Majumdar, Phys. Rev. E 77, 041108 (2008) ADSMathSciNetCrossRefGoogle Scholar
  105. 105.
    S.N. Majumdar, M. Vergassola, Phys. Rev. Lett. 102, 060601 (2009) ADSCrossRefGoogle Scholar
  106. 106.
    C. Monthus, arXiv:1904.02448
  107. 107.
    D. Andrieux, P. Gaspard, J. Stat. Mech. 2008, P11007 (2008) CrossRefGoogle Scholar
  108. 108.
    C. Maes, K.l Netocny, B. Wynants, J. Phys. A: Math. Theor. 42, 365002 (2009) ADSCrossRefGoogle Scholar
  109. 109.
    M. Mariani, L. Zambotti, Adv. Appl. Probab. 48, 648 (2016) CrossRefGoogle Scholar
  110. 110.
    A. Faggionato, arXiv:1709.05653

Copyright information

© EDP Sciences / Società Italiana di Fisica / Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institut de Physique Théorique, Université Paris Saclay, CNRS, CEAGif-sur-YvetteFrance

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