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Stochastic thermodynamics of holonomic systems

  • Stefano GiordanoEmail author
Regular Article
  • 19 Downloads

Abstract

Stochastic thermodynamics is a recently introduced approach to deals with small systems in contact with one or more thermal baths. This theory has been applied to systems of unconstrained particles to investigate the role of the thermodynamics principles in micro- and nano-scale systems and to demonstrate some important fluctuations theorems. Nowadays, the manipulations of small systems with advanced nanotechnologies provided the experimental evidence of most of results based on stochastic thermodynamics. Here, this approach is generalized to consider arbitrary holonomic systems subjected to arbitrary external forces and described by Lagrange and Hamilton equations of motion. In both the underdamped and overdamped cases, the principles of thermodynamics are obtained in the out-of-equilibrium regime by giving microscopic interpretations of heat, energy and entropy. To do this, the Klein-Kramers (for the underdamped case) and Smoluchowski (for the overdamped case) equations are used in covariant form to be consistent with the Brownian motion on smooth manifolds. Moreover, explicit expressions for the entropy production have been obtained and can be applied to the non-equilibrium thermodynamics of holonomic systems.

Graphical abstract

Keywords

Statistical and Nonlinear Physics 

References

  1. 1.
    R. Brown, Philos. Mag. 4, 161 (1828) CrossRefGoogle Scholar
  2. 2.
    R. Brown, Philos. Mag. 6, 161 (1829) CrossRefGoogle Scholar
  3. 3.
    A. Fick, Ann. Phys. (Leipzig) 19, 371 (1855) Google Scholar
  4. 4.
    A. Einstein, Ann. Phys. (Leipzig) 17, 549 (1905) CrossRefGoogle Scholar
  5. 5.
    A. Einstein, Ann. Phys. (Leipzig) 19, 371 (1906) CrossRefGoogle Scholar
  6. 6.
    M. von Smoluchowski, Ann. Phys. (Leipzig) 21, 756 (1906) CrossRefGoogle Scholar
  7. 7.
    P. Langevin, C. R. Acad. Sci. (France) 146, 530 (1908) Google Scholar
  8. 8.
    J.B. Perrin, C. R. Acad. Sci. (France) 158, 1168 (1914) Google Scholar
  9. 9.
    J.B. Perrin, Discontinuous Structure of Matter, Nobel Lecture, 1926 Google Scholar
  10. 10.
    A.D. Fokker, Ann. Phys. 348, 810 (1914) CrossRefGoogle Scholar
  11. 11.
    M. Planck, Sitzungsber. Preuss. Akad. Wiss. 324, 142 (1917) Google Scholar
  12. 12.
    A. Kolmogoroff, Math. Ann. 104, 415 (1931) MathSciNetCrossRefGoogle Scholar
  13. 13.
    O. Klein, Arkiv für Matematik, Astronomi och Fysik 16, 1 (1921) Google Scholar
  14. 14.
    H.A. Kramers, Physica 7, 284 (1940) MathSciNetCrossRefGoogle Scholar
  15. 15.
    S. Chandrasekhar, Rev. Mod. Phys. 15, 1 (1943) CrossRefGoogle Scholar
  16. 16.
    W. Coffey, Adv. Chem. Phys. 63, 69 (1985) Google Scholar
  17. 17.
    N.G. van Kampen, Stochastic Processes in Physics and Chemistry (Elsevier, Amsterdam, 1981) Google Scholar
  18. 18.
    H. Risken, The Fokker-Planck equation (Springer Verlag, Berlin, 1989) Google Scholar
  19. 19.
    W.T. Coffey, Yu.P. Kalmykov, J.P. Waldron, The Langevin equation (World Scientific, Singapore, 2004) Google Scholar
  20. 20.
    K. Sekimoto, J. Phys. Soc. Jpn. 66, 1234 (1997) CrossRefGoogle Scholar
  21. 21.
    K. Sekimoto, Stochastic Energetics (Springer, Berlin, 2010) Google Scholar
  22. 22.
    U. Seifert, Phys. Rev. Lett. 95, 040602 (2005) CrossRefGoogle Scholar
  23. 23.
    U. Seifert, Eur. Phys. J. B 64, 423 (2008) CrossRefGoogle Scholar
  24. 24.
    U. Seifert, Rep. Prog. Phys. 75, 126001 (2012) CrossRefGoogle Scholar
  25. 25.
    J. Schnakenberg, Rev. Mod. Phys. 48, 571 (1976) MathSciNetCrossRefGoogle Scholar
  26. 26.
    M. Esposito, C. Van den Broeck, Phys. Rev. E 82, 011143 (2010) CrossRefGoogle Scholar
  27. 27.
    C. Van den Broeck, M. Esposito, Phys. Rev. E 82, 011144 (2010) CrossRefGoogle Scholar
  28. 28.
    T. Tomé, M.J. de Oliveira, Phys. Rev. E 82, 021120 (2010) CrossRefGoogle Scholar
  29. 29.
    T. Tomé, M.J. de Oliveira, Phys. Rev. Lett. 108, 020601 (2012) CrossRefGoogle Scholar
  30. 30.
    T. Tomé, M.J. de Oliveira, Phys. Rev. E 91, 042140 (2015) MathSciNetCrossRefGoogle Scholar
  31. 31.
    C. Jarzynski, Phys. Rev. Lett. 78, 2690 (1997) CrossRefGoogle Scholar
  32. 32.
    C. Jarzynski, Phys. Rev. E 56, 5018 (1997) CrossRefGoogle Scholar
  33. 33.
    G. Crooks, Phys. Rev. E 60, 2721 (1999) CrossRefGoogle Scholar
  34. 34.
    D. Collin, F. Ritort, C. Jarzynski, S.B. Smith, I. Tinoco, C. Bustamante, Nature 437, 231 (2005) CrossRefGoogle Scholar
  35. 35.
    C. Jarzynski, C. R. Phys. 8, 495 (2007) CrossRefGoogle Scholar
  36. 36.
    M. Esposito, U. Harbola, S. Mukamel, Rev. Mod. Phys. 81, 1665 (2009) CrossRefGoogle Scholar
  37. 37.
    P. Caldirola, L.A. Lugiato, Physica A 116, 248 (1982) MathSciNetCrossRefGoogle Scholar
  38. 38.
    A.O. Caldeira, A.J. Leggett, Physica A 121, 587 (1983) MathSciNetCrossRefGoogle Scholar
  39. 39.
    B. Bianco, E. Moggia, S. Giordano, W. Rocchia, A. Chiabrera, Il Nuovo Cimento 116, 155 (2001) Google Scholar
  40. 40.
    P. Talkner, E. Lutz, P. Hänggi, Phys. Rev. E 75, 050102(R) (2007) CrossRefGoogle Scholar
  41. 41.
    M. Campisi, P. Talkner, P. Hänggi, Phys. Rev. Lett. 102, 210401 (2009) CrossRefGoogle Scholar
  42. 42.
    G. Watanabe, P. Venkatesh, P. Talkner, M. Campisi, P. Hänggi, Phys. Rev. E 89, 032114 (2014) CrossRefGoogle Scholar
  43. 43.
    S.M. Smith, Y. Cui, C. Bustamante, Science 271, 795 (1996) CrossRefGoogle Scholar
  44. 44.
    J.F. Marko, E.D. Siggia, Macromolecules 28, 8759 (1995) CrossRefGoogle Scholar
  45. 45.
    F. Ritort, J. Phys.: Condens. Matter 18, R531 (2006) Google Scholar
  46. 46.
    F. Manca, S. Giordano, P.L. Palla, R. Zucca, F. Cleri, L. Colombo, J. Chem. Phys. 136, 154906 (2012) CrossRefGoogle Scholar
  47. 47.
    F. Manca, S. Giordano, P.L. Palla, F. Cleri, L. Colombo, J. Chem. Phys. 137, 244907 (2012) CrossRefGoogle Scholar
  48. 48.
    M. Rief, J.M. Fernandez, H.E. Gaub, Phys. Rev. Lett. 81, 4764 (1998) CrossRefGoogle Scholar
  49. 49.
    F. Manca, S. Giordano, P.L. Palla, F. Cleri, L. Colombo, Phys. Rev. E 87, 032705 (2013) CrossRefGoogle Scholar
  50. 50.
    M. Benedito, S. Giordano, J. Chem. Phys. 149, 054901 (2018) CrossRefGoogle Scholar
  51. 51.
    M. Benedito, S. Giordano, Phys. Rev. E 98, 052146 (2018) CrossRefGoogle Scholar
  52. 52.
    M. Benedito, F. Manca, S. Giordano, Inventions 4, 19 (2019) CrossRefGoogle Scholar
  53. 53.
    R.G. Winkler, Soft Matter 6, 6183 (2010) CrossRefGoogle Scholar
  54. 54.
    F. Manca, S. Giordano, P.L. Palla, F. Cleri, Physica A 395, 154 (2014) MathSciNetCrossRefGoogle Scholar
  55. 55.
    K. Svoboda, C. Schmidt, B. Schnapp, S. Block, Nature 365, 721 (1993) CrossRefGoogle Scholar
  56. 56.
    M. Magnasco, Phys. Rev. Lett. 71, 1477 (1993) CrossRefGoogle Scholar
  57. 57.
    P. Hänggi, F. Marchesoni, Rev. Mod. Phys. 81, 387 (2009) CrossRefGoogle Scholar
  58. 58.
    R. Perez-Carrasco, J.M. Sancho, Phys. Rev. E 84, 041915 (2011) CrossRefGoogle Scholar
  59. 59.
    L. Landau, E. Lifshitz, Phys. Z. Sowjetunion 8, 153 (1935) Google Scholar
  60. 60.
    T.L. Gilbert, Phys. Rev. 100, 1243 (1955) (abstract only) Google Scholar
  61. 61.
    T.L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004) CrossRefGoogle Scholar
  62. 62.
    W.F. Brown, J. Appl. Phys. 30, S130 (1959) CrossRefGoogle Scholar
  63. 63.
    W.F. Brown, IEEE Trans. Magn. 15, 1196 (1979) CrossRefGoogle Scholar
  64. 64.
    G. Bertotti, I. Mayergoyz, C. Serpico, Nonlinear Magnetization Dynamic in Nanosystems (Elsevier, Oxford, 2000) Google Scholar
  65. 65.
    D.R. Fredkin, Physica B 306, 26 (2001) CrossRefGoogle Scholar
  66. 66.
    S.I. Denisov, K. Sakmann, P. Talkner, P. Hänggi, Phys. Rev. B 75, 184432 (2007) CrossRefGoogle Scholar
  67. 67.
    P.M. Déjardin, D.S.F. Crothers, W.T. Coffey, D.J. McCarthy, Phys. Rev. E 63, 021102 (2001) CrossRefGoogle Scholar
  68. 68.
    S. Giordano, Y. Dusch, N. Tiercelin, P. Pernod, V. Preobrazhensky, Eur. Phys. J. B 86, 249 (2013) CrossRefGoogle Scholar
  69. 69.
    S. Giordano, Y. Dusch, N. Tiercelin, P. Pernod, V. Preobrazhensky, J. Phys. D: Appl. Phys. 46, 325002 (2013) CrossRefGoogle Scholar
  70. 70.
    N. Tiercelin, Y. Dusch, S. Giordano, A. Klimov, V. Preobrazhensky, P. Pernod, in Nanomagnetic and Spintronic Devices for Energy-Efficient Memory and Computing, edited by S. Bandyopadhyay, J. Atulasimha (John Wiley & Sons Ltd, Chichester, 2016), Chap. 8 Google Scholar
  71. 71.
    R. Pan, T.M. Hoang, Z. Fei, T. Qiu, J. Ahn, T. Li, H.T. Quan, Phys. Rev. E 98, 052105 (2018) CrossRefGoogle Scholar
  72. 72.
    Y. Murashita, M. Esposito, Phys. Rev. E 94, 062148 (2016) CrossRefGoogle Scholar
  73. 73.
    F. Manca, P.-M. Déjardin, S. Giordano, Ann. Phys. (Berlin) 528, 381 (2016) CrossRefGoogle Scholar
  74. 74.
    M. Fixman, Proc. Natl. Acad. Sci. USA 71, 3050 (1974) CrossRefGoogle Scholar
  75. 75.
    G. Ciccotti, M. Ferrario, in Classical and Quantum Dynamics in Condensed Phase Simulations, edited by B.J. Berne, G. Ciccotti, D.F. Coker (World Scientific, Singapore, 1998), Chap. 4 Google Scholar
  76. 76.
    G. Ciccotti, M. Ferrario, Computation 6, 11 (2018) CrossRefGoogle Scholar
  77. 77.
    F. Gantmacher, Lectures in Analytical Mechanics (MIR Publisher, Moscow, 1975) Google Scholar
  78. 78.
    K. Itô, Nagoya Math. J. 1, 35 (1950) MathSciNetCrossRefGoogle Scholar
  79. 79.
    R.L. Stratonovich, SIAM J. Control Optim. 4, 362 (1966) CrossRefGoogle Scholar
  80. 80.
    P. Hänggi, H. Thomas, Phys. Rep. 88, 207 (1982) MathSciNetCrossRefGoogle Scholar
  81. 81.
    Yu.L. Klimontovich, Statistical Theory of Open Systems (Kluver Academic, Dordrecht, 1995) Google Scholar
  82. 82.
    I.M. Sokolov, Chem. Phys. 375, 359 (2010) CrossRefGoogle Scholar
  83. 83.
    S.I. Denisov, A.N. Vitrenko, W. Horsthemke, Phys. Rev. E 68, 046132 (2003) CrossRefGoogle Scholar
  84. 84.
    V. Méndez, S.I. Denisov, D. Campos, W. Horsthemke, Phys. Rev. E 90, 012116 (2014) CrossRefGoogle Scholar
  85. 85.
    J.W. Gibbs, Elementary principles in statistical mechanics (Charles Scribner’s Sons, New York, 1902) Google Scholar
  86. 86.
    H. White, Bull. Math. Biophys. 27, 135 (1965) CrossRefGoogle Scholar
  87. 87.
    V. Hnizdo, M.K. Gilson, Entropy 12, 578 (2010) MathSciNetCrossRefGoogle Scholar
  88. 88.
    M. Polettini, J. Stat. Mech.: Theory Exp. 2013, P07005 (2013) MathSciNetCrossRefGoogle Scholar
  89. 89.
    M.P. do Carmo Differential Geometry of Curves and Surfaces (Prentice-Hall, New York, 1976) Google Scholar
  90. 90.
    T. Levi-Civita, The absolute differential calculus (Blackie & Son Limited, London, 1946) Google Scholar
  91. 91.
    P. Castro-Villarreal, J. Stat. Mech.: Theory Exp. 2010, P08006 (2010) CrossRefGoogle Scholar
  92. 92.
    P. Castro-Villarreal, J. Stat. Mech.: Theory Exp. 2014, P05017 (2014) MathSciNetCrossRefGoogle Scholar
  93. 93.
    P. Castro-Villarreal, A. Villada-Balbuena, J.M. Méndez-Alcaraz, R. Castañeda-Priego, S. Estrada-Jiménez, J. Chem. Phys. 140, 214115 (2014) CrossRefGoogle Scholar
  94. 94.
    H. Kleinert, S.V. Shabanov, J. Phys. A: Math. Gen. 31, 7005 (1998) CrossRefGoogle Scholar
  95. 95.
    Z. Schuss, Theory and Applications of Stochastic Differential Equations (John Wiley & Sons, New York, 1980) Google Scholar
  96. 96.
    Z. Schuss, Theory and Applications of Stochastic Processes (Springer, New York, 2010) Google Scholar
  97. 97.
    P. Hsu, Contemp. Math. AMS 73, 95 (1988) CrossRefGoogle Scholar
  98. 98.
    K. Itô, in Proc. Internat. Congr. Math. (Stockholm, Inst. Mittag-Leffler, Djursholm, 1962), p. 536 Google Scholar
  99. 99.
    W.S. Kendall, Acta Appl. Math. 9, 29 (1987) MathSciNetCrossRefGoogle Scholar
  100. 100.
    D.W. Stroock, Bull. Amer. Math. Soc. 33, 135 (1996) MathSciNetCrossRefGoogle Scholar
  101. 101.
    R. Graham, Z. Phys. B 26, 397 (1977) MathSciNetCrossRefGoogle Scholar
  102. 102.
    H. Grabert, M.S. Green, Phys. Rev. A 19, 1747 (1979) MathSciNetCrossRefGoogle Scholar
  103. 103.
    H. Grabert, R. Graham, M.S. Green, Phys. Rev. A 21, 2136 (1980) MathSciNetCrossRefGoogle Scholar
  104. 104.
    G.E. Uhlenbeck, L.S. Ornstein, Phys. Rev. 36, 823 (1930) CrossRefGoogle Scholar
  105. 105.
    M.C. Wang, G.E. Uhlenbeck, Rev. Mod. Phys. 17, 323 (1945) CrossRefGoogle Scholar
  106. 106.
    N.G. van Kampen, J. Stat. Phys. 44, 1 (1986) MathSciNetCrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Institute of Electronics, Microelectronics and Nanotechnology – UMR 8520, Univ. Lille, CNRS, Centrale Lille, ISEN, Univ. Valenciennes, LIA LICS/LEMACLilleFrance

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