Skip to main content
SpringerLink
Log in

Non-equilibrium statistical mechanics of complex systems: An overview

  • Published:
La Rivista del Nuovo Cimento Aims and scope

    We’re sorry, something doesn't seem to be working properly.

    Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Summary

The ensemble formalism has been extremely successful in the handling of the grandiose theoretical scheme of statistical mechanics and thermodynamics initiated by Maxwell, Boltzmann and Gibbs, which has been given concrete and consistent foundations to the study of the many situations present in condensedmatter physics. However, its use is hampered when dealing with certain complex phenomena for which the researcher may not have an access to the information on all the constraints relevant to the problem in hands (so-called hidden constraints), which leads to poor predictions. In an attempt to improve predictions there have been introduced, beginning in the past 1950s, and pioneered by P. Levy in the 1930s, auxiliary approaches which attempt to assuage the difficulty, but at the price of not being fully consistent and depending on free parameters. This is done in the framework of the variational (extremum principle) approach in statistical mechanics founded on information theory. In it, the general and well-established Boltzmann-Gibbs canonical scheme follows from maximization with given constraints of Gibbs-Boltzmann-Shannon information-theoretic entropy (better called measure of uncertainty of information): it is considered to be the only consistent probability measure of information. The other (say non-canonical or heterotypical) auxiliary approaches are based on replacing GBS information-theoretic entropy by others, which are used to derive non-conventional probability distributions for non-equilibrium systems. We present a detailed description of their construction and a clarification of their scope, interpretation and utility. Also, resorting to the particular case of Renyi’s approach the construction of a non-equilibrium ensemble formalism is described. The non-conventional distribution functions of fermions and bosons are discussed. The use of the formalism is illustrated via the analysis of experimental results in the case of fractal-like structured systems. Also a purely theoretical analysis is done in the cases of an ideal gas and of radiation comparing the conventional and non-conventional approaches. In all of these situations it is discussed which are the difficulties involved (hidden constraints in an insufficient description) which require to resort to the non-conventional approach, and what determines the value of the parameter(s) that the formalism introduces in each case, namely, its (their) dependence on the system characteristics and the experimental protocol.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Montroll E. W. and Shlesinger M. F., J. Stat. Phys., 12 (1983) 209.

  2. Fisher R. A., Philos. Trans. R. Soc. London A, 222 (1922) 309.

    Article  ADS  Google Scholar 

  3. Balian R., Am.J.Phys., 67 (1999) 1078.

    Article  ADS  Google Scholar 

  4. Landsberg P. T., Braz. J. Phys., 29 (1999) 46.

    Article  ADS  Google Scholar 

  5. Zaslavsky G. M., Phys. Rep., 371 (2002) 461.

    Article  ADS  MathSciNet  Google Scholar 

  6. Shlesinger M. F., Zaslavsky G. M. and Frish U., Levy Flights and Related Topics in Physics (Springer, Berlin, Germany) 1985.

    Google Scholar 

  7. Ebeling W., Statistical Physics and Thermodynamics of Nonlinear Non-equilibrium Systems (World Scientific, Singapore) 1993.

    Book  Google Scholar 

  8. Feistel R. and Ebeling W., Evolution of Complex Systems (Kluwer Academic, Dordrecht, The Netherlands) 1989.

    MATH  Google Scholar 

  9. Ebeling W., Physica A, 182 (1992) 108.

    Article  ADS  Google Scholar 

  10. Beck C. and Cohen E. G. D., arXiv.org/cond-mat/0205097 (2002).

    Google Scholar 

  11. Havrda J. and Charvat F., Kybernetica, 3 (1967) 30.

    Google Scholar 

  12. Renyi A., Proceedings of the 4th Berkeley Symposium Math. Stat. Prob., 1 (1961) 547.

    Google Scholar 

  13. Takens F. and Verbitski E., Nonlinearity, 11 (1998) 771.

    Article  ADS  MathSciNet  Google Scholar 

  14. Grassberger P. and Procaccia I., Phys. Rev. A, 28 (1983) 2591.

    Article  ADS  Google Scholar 

  15. Grassberger P., Physica D, 14 (1985) 365.

    Article  ADS  MathSciNet  Google Scholar 

  16. Jizba P. and Arimitzu T., Ann. Phys., 312 (2004) 17 and also arXiv.org/cond-mat/0207707 and 0307698.

    Article  ADS  Google Scholar 

  17. Sharma B. D. and Mittal D. P., J. Math. Sci., 10 (1975) 28.

    MathSciNet  Google Scholar 

  18. Vasyliunas V. M., J. Geophys. Res., 73 (1968) 2839.

    Article  ADS  Google Scholar 

  19. Kaniadakis G., Phys. Rev. E, 66 (2002) 56125.

    Article  ADS  MathSciNet  Google Scholar 

  20. Luzzi R., Vasconcellos A. R. and Ramos J. G., Predictive Statistical Mechanics: A Non-equilibrium Ensemble Formalism (Kluwer Academic, Dordrecht, The Netherlands) 2002.

    Book  MATH  Google Scholar 

  21. Jaynes E. T.,in E. T. Jaynes Papers on Probability, Statistics, and Statistical Physics, edited by Rosenkrantz R. D. (Reidel-Kluwer Academic, Dordrecht, The Netherlands) 1983.

  22. Jaynes E. T.,in Frontiers of Non-equilibrium Statistical Physics, edited by Moore g. t. and Scully m. ο. (Plenum, New York, USA) 1986, pp. 33–55.

  23. Shannon C. E. and Weaver W., The Mathematical Theory of Communication (Univ. Illinois Press, Urbana, USA) 1948.

    MATH  Google Scholar 

  24. Cox R. T., The Algebra of Probable Inference (John Hopkings Press, Baltimore, USA) 1961.

    MATH  Google Scholar 

  25. Jaynes E. T., Probability Theory: The Logic of Science (Cambridge Univ. Press, Cambridge, UK, 2002).

    Google Scholar 

  26. Cho A., Science, 297 1268 (2002).

    Article  Google Scholar 

  27. Kapur J. N. and Kesavan H. K., Entropy Optimization Principles with Applications (Academic, Boston, USA) 1992.

    Book  Google Scholar 

  28. Luzzi R., Vasconcellos A. R. and Ramos J. G., Science, 298 (2002) 1171.

    Article  Google Scholar 

  29. Grassberger P., Phys. Rev. Lett., 95 (2005) 140601, and also arXiv.org/cond-mat/0508110.

    Article  ADS  Google Scholar 

  30. Tsallis C., Braz. J. Phys., 29 (1999) 1.

    Article  ADS  Google Scholar 

  31. Brillouin L.,in Science and Information Theory (Academic, New York, USA) 1956.

    Book  MATH  Google Scholar 

  32. Nauenberg M., Phys. Rev. E, 67 (2003) 036114; 69 (2004) 038102.

    Article  ADS  Google Scholar 

  33. Balian R. and Nauenberg N., Europhysics News, 37 (2006) 9.

    Google Scholar 

  34. Luzzi R., Vasconcellos A. R. and Ramos J. G., Europhysics News, 37 (2006) 11.

    Article  Google Scholar 

  35. Balian R., Entropy, a Protean Concept,in Poincare Seminar 2003 (Birkhauser, Basel, Swiss) 2004, pp. 119–144.

    Book  Google Scholar 

  36. Pathria R. K., Statistical Mechanics (Butterworth-Heinemann, Boston, USA) 1996.

    MATH  Google Scholar 

  37. Sklar L., Theory and Truth: Philosophical Critique Within Foundational Science (Oxford University Press, Oxford, UK) 2000.

    Google Scholar 

  38. Hill T., J. Chem. Phys., 36 (1962) 3182.

    Article  ADS  Google Scholar 

  39. Kulback S. and Leibler R. A., Ann. Math. Stat., 22 (1951) 79.

    Article  Google Scholar 

  40. Jaynes E. T., Am. J. Phys., 33 (1965) 391.

    Article  ADS  Google Scholar 

  41. Balian R. and Balazs N. L., Ann. Phys., 179 (1987) 97.

    Article  ADS  Google Scholar 

  42. Grandy W. T., Principles of Statistical Mechanics: Equilibrium Theory (Reidel-Kluwer Academic, Dordrecht, The Netherlands) 1987.

    Book  MATH  Google Scholar 

  43. Grandy W. T., Principles of Statistical Mechanics: Non-equilibrium Phenomena (Reidel-Kluwer Academic, Dordrecht, The Netherlands) 1988.

    MATH  Google Scholar 

  44. Ochs W., Rep. Math. Phys., 8 (1975) 109.

    Article  ADS  MathSciNet  Google Scholar 

  45. Landsberg P. T. and Vedral V., Phys. Lett. A, 247 (1998) 211.

    Article  ADS  MathSciNet  Google Scholar 

  46. Luzzi R., Vasconcellos A. R. and Ramos J. G., The Theory of Irreversible Processes: Non-equilibrium Statistical Ensemble Formalism, Notas de Flsica IFGW, Vol. 27 (Unicamp Press, Campinas, SP, Brazil) 2005; Riv. Nuovo Cimento, 29 (2) (2006) 1-85.

  47. Laplace P. S., Essay Philosophique sur les Probabilities (Bachelier, Paris, France) 1825: English translation: A Philosophical Essay on Probability (Reprint by Dover, New York, USA) 1995.

    Google Scholar 

  48. Jaynes E. T.,in The Maximum Entropy Formalism, edited by Tribus m. and Levine r. d. (MIT Press, Cambridge, MA, USA) 1978, pp. 15–118.

  49. Onsager L., Phys. Rev., 37 (1931) 405.

    Article  ADS  Google Scholar 

  50. Casimir H. G. B., Rev. Mod. Phys., 17 (1945) 343.

    Article  ADS  Google Scholar 

  51. Jaynes E. T., Phys. Rev., 106 (1957) 620.

    Article  ADS  MathSciNet  Google Scholar 

  52. Jaynes E. T., Phys. Rev., 108 (1957) 171.

    Article  ADS  MathSciNet  Google Scholar 

  53. Jeffreys H., Probability Theory (Clarendon, Oxford, UK) 1961.

    MATH  Google Scholar 

  54. Jeffreys H., Scientific Inference (Cambridge Univ. Press, Cambridge, UK) 1973.

    MATH  Google Scholar 

  55. Luzzi R., Vasconcellos A. R. and Ramos J. G., J. Mod. Phys. B, 14 (2000) 3189; Fortschr. Phys. Prog. Phys., 38 (1990) 887.

    Article  ADS  Google Scholar 

  56. Uffink J., Stud. Hist. Philos. Mod. Phys., 26 (1995) 223.

    Article  MathSciNet  Google Scholar 

  57. Elssasser W., Phys. Rev., 52 (1937) 987.

    Article  ADS  Google Scholar 

  58. Landauer R., Phys. Rev. A, 12 (1975) 636.

    Article  ADS  Google Scholar 

  59. Anderson P. W., Phys. Today, 45(1) (1992) 9.

    Article  ADS  Google Scholar 

  60. Heisenberg W., The Physical Conception of Nature (Hutchinson, London, UK) 1958.

    MATH  Google Scholar 

  61. Born M., Experiment and Theory in Physics (Dover, New York, USA) 1956.

    MATH  Google Scholar 

  62. Hawkings S., 1990—Yearbook of Science and Future (Encyclopaedia Britannica, Chicago, USA) 1989.

    Google Scholar 

  63. Sklar L., Physics and Chance: Philosophical Issues in the Foundations of Statistical Mechanics (Cambridge Univ. Press, Cambridge, UK) 1993.

    Book  Google Scholar 

  64. VAN Fraassen B. C., Br. J. Philos. Sci., 32 (1981) 375.

    Article  Google Scholar 

  65. Fraassen B. C., Hughes R. I. and Herman G., Br. Philos. Sci., 37 (1986) 453.

    Article  Google Scholar 

  66. Fraassen B. C., Laws and Symmetry (Claredon, Oxford, UK) 1989.

    Book  Google Scholar 

  67. Herniter J. D., J. Market Res., 10 (1973) 361.

    Article  Google Scholar 

  68. Zubarev D. N., Morozov V. N. and G. Röpke, Statistical Mechanics of Non-equilibrium Processes: Basic Concepts, Kinetic Theory (Akademie Verlag Wiley-VHC, Berlin, Germany) 1996.

    MATH  Google Scholar 

  69. Hobson A., J. Chem. Phys., 45 (1966) 1352.

    Article  ADS  Google Scholar 

  70. Hobson A., Am. J. Phys., 34 (1966) 411.

    Article  ADS  Google Scholar 

  71. Garcia-Colin L. S., Vasconcellos A. R. and Luzzi R., J. Non-Equilib. Thermodyn., 19 (1994) 24.

    Article  ADS  Google Scholar 

  72. Luzzi R., Vasconcellos A. R. and Ramos J. G., Statistical Foundations of Irreversible Thermodynamics (Teubner-Bertelsmann Springer, Stuttgart, Germany) 2000.

    Book  MATH  Google Scholar 

  73. Robertson B., Phys. Rev., 144 (1966) 151; 160 (1967) 175.

    Article  ADS  MathSciNet  Google Scholar 

  74. Lauck L., Vasconcellos A. R. and Luzzi R., Physica A, 168 (1990) 789.

    Article  ADS  Google Scholar 

  75. Luzzi R. and Vasconcellos A. R., J. Stat. Phys., 23 (1980) 539.

    Article  ADS  Google Scholar 

  76. Algarte A. C., Vasconcellos A. R. and Luzzi R., Phys. Stat. Sol. (b), 173 (1992) 487.

    Article  ADS  Google Scholar 

  77. Mesquita M. V., Vasconcellos A. R. and Luzzi R., Phys. Rev. Lett., 80 (1998) 2008.

    Article  ADS  Google Scholar 

  78. Fonseca A. F., Mesquita M. V., Vasconcellos A. R. and Luzzi R., J. Chem. Phys., 112 (2000) 3967.

    Article  ADS  Google Scholar 

  79. Rodrigues C. G., Vasconcellos A. R. and Luzzi R., Solid State Commun., 140 (2006) 435.

    Article  Google Scholar 

  80. Ramos J. G., Vasconcellos A. R. and Luzzi R., Fortschr. Phys./Prog. Phys., 43 (1995) 265.

    Article  ADS  Google Scholar 

  81. Dedeurwaerdene R., Casas-Vazquez J., Jou D. and Lebon G., Phys. Rev. E, 53 (1996) 498.

    Article  ADS  Google Scholar 

  82. Madureira J. R., Vasconcellos A. R., Luzzi R., Casas-Vazquez J. and Jou D., J. Chem. Phys., 108 (1998) 7568.

    Article  ADS  Google Scholar 

  83. Ramos J. G., Vasconcellos A. R. and Luzzi R., A Generalized Non-Classical Navier-Stokes Equation in a Nonequilibrium Ensemble Formalism, IFGW-Unicamp Internal Reports (2006), and future publication.

    Google Scholar 

  84. Jou D., Casas-Vazquez J., Vasconcellos A. R., Madureira J. R. and Luzzi R., J. Chem. Phys., 116 (2002) 1571.

    Article  ADS  Google Scholar 

  85. Vasconcellos A. R., Ramos J. G., Luzzi R., Silva A. A. P., Jou D. and Casas-Vazquez J., Hydrodynamic Motion in Complex-Structured Materials I, IFGW-Unicamp Internal Reports (2006), and future publication.

    Google Scholar 

  86. Vasconcellos A. R., Ramos J. G., Luzzi R., Silva A. A. P., Jou D. and Casas-Vazquez J., Hydrodynamic Motion in Complex-Structured Materials II, IFGW-Unicamp Internal Reports (2006), and future publication.

  87. Gyftopoulos E. P. and Cubukçu E., Phys. Rev. E, 55 (1997) 3851.

    Article  ADS  MathSciNet  Google Scholar 

  88. Gyftopoulos E., Physica A, 307 (2002) 405.

    Article  ADS  MathSciNet  Google Scholar 

  89. Csiszer I., Periodic Math. Hungarica, 9 (1972) 191.

    Article  Google Scholar 

  90. Hentschel H. G. and Procaccia I., Physica D, 8 (1983) 435.

    Article  ADS  MathSciNet  Google Scholar 

  91. Kapur J. N., The Mathematics Seminar, 4 (1967) 78.

    Google Scholar 

  92. Prigogine I., From Being to Becoming (Freeman, San Francisco, USA) 1980.

    Google Scholar 

  93. Beck C. and Schlögl F., Thermodynamics of Chaotic Systems (Cambridge Univ. Press, Cambridge, UK) 1993.

    Book  MATH  Google Scholar 

  94. Renyi A., Probability Theory (North Holland, Amsterdam, The Netherlands) 1970.

    MATH  Google Scholar 

  95. Penrose Ο., Rep. Prog. Phys., 42 (1979) 1938.

    Article  ADS  Google Scholar 

  96. Kirkwood J. G., J. Chem. Phys., 14 (1946) 180.

    Article  ADS  Google Scholar 

  97. Peierls R., Lect. Notes Phys., Vol. 31 (Springer, Berlin, Germany) 1974.

  98. Tischenko S. V., Theor. Math. Phys., 25 (1975) 1218.

    Article  Google Scholar 

  99. Hassan S. A., Vasconcellos A. R. and Luzzi R., Physica A, 235 (1997) 345.

    Article  ADS  Google Scholar 

  100. Luzzi R., Vasconcellos A. R. and Ramos J. G., Riv. Nuovo Cimento, 24(3) (2001) 1–70.

    Google Scholar 

  101. Bogoliubov N. N., Lectures in Quantum Statistics II (Gordon and Breach, New York, USA) 1970.

    Google Scholar 

  102. Gell-Mann M. and Goldberger M. L., Phys. Rev., 91 (1953) 398.

    Article  ADS  MathSciNet  Google Scholar 

  103. Krylov N. S., Works on the Foundations of Statistical Mechanics (Princeton Univ. Press, Princeton, USA) 1979.

    Google Scholar 

  104. Madureira J., Vasconcellos A. and Luzzi R., J. Chem. Phys., 109 (1998) 2099.

    Article  ADS  Google Scholar 

  105. Luzzi R., Vasconcellos A. R., Jou D. and Casas-Vazquez J., J. Chem. Phys., 107 (1997) 7383.

    Article  ADS  Google Scholar 

  106. Nettleton R. E., Can. J. Phys., 72 (1994) 106.

    Article  ADS  Google Scholar 

  107. Singh J., Properties of Semiconductors and their Heterostructures (McGraw-Hill, London, UK) 1993.

    Google Scholar 

  108. Family F. and Viczek T., Dynamics of Fractal Surfaces (World Scientific, Singapore) 1991.

    Book  Google Scholar 

  109. Luzzi R., Vasconcellos A. R., Casas-Vazquez J. and Jou D., Physica A, 248 (1997) 111.

    Article  ADS  Google Scholar 

  110. Jou D. and Casas-Vazquez J., Rep. Prog. Phys., 66 (2003) 1937.

    Article  ADS  Google Scholar 

  111. Laureto E., Vasconcellos A. R., Meneses E. A. and Luzzi R., J. Mod. Phys. B, 18 (2004) 1743; Chaos, Solitons & Fractals, 28 (2006) 8.

    Article  ADS  Google Scholar 

  112. Runge E. and Zimmermann R., Adv. Solid State Phys., 38 (1998) 251.

    Article  Google Scholar 

  113. Singh J. and Bajaj K. K., J. Appl. Phys., 57 (1985) 5433.

    Article  ADS  Google Scholar 

  114. Borri P., Gurioli M., Colocci M., Marteli F., M. Capizzi, Patane A. and Polimeni A., J. Appl. Phys., 80 (1996) 3011.

    Article  ADS  Google Scholar 

  115. Fujiwara K., Kanamoto K. and Tsukuda N., Phys. Rev. B, 40 (1989) 9698.

    Article  ADS  Google Scholar 

  116. Jahn U., Khwok S. H., Ramsteiner M., Hey R., Grahn H. T. and Runge E., Phys. Rev. B, 54 (1996) 2733.

    Article  ADS  Google Scholar 

  117. Vasconcellos A. R., Brasil M. J., Silva A. A. P., Luzzi R., Kinetic-Hydrodynamic Approach for Plasma Fluid in Semiconductors, IFGW-Unicamp Internal Report (2006), and future publication.

    Google Scholar 

  118. Vasconcellos A. R., Ramos J. G., Gorenstein A., Kleinke M. U., Cruz T. G. S. and Luzzi R., J. Mod. Phys. B, 20 (2006) 4821.

    Article  ADS  Google Scholar 

  119. Julien C. and Nazri G. A., Solid-State Batteries (Kluwer, Dordrecht, The Netherlands) 1994.

    Google Scholar 

  120. Pajkossy T. and Nyikos L., Electrochem. Acta, 14 (1989) 181.

    Article  Google Scholar 

  121. Dotto M. E. R. and Kleinke M. U., Physica A, 295 (2001) 149.

    Article  ADS  Google Scholar 

  122. Crank J., The Mathematics of Diffusion (Oxford Univ. Press, Oxford, UK) 1975.

    MATH  Google Scholar 

  123. Jou D., Casas-Vazquez J., Madureira J. R., Vasconcellos A. R. and Luzzi R., J. Chem. Phys., 116 (2002) 1561.

    Article  ADS  Google Scholar 

  124. Vasconcellos A. R., Algarte A. C. and Luzzi R., Phys. Rev. B, 48 (1993) 10873.

    Article  ADS  Google Scholar 

  125. Jou D., Casas-Vazquez J. and Criado-Sancho M., Thermodynamics of Fluids Under Flow (Springer, Berlin, Germany) 2001.

    Book  MATH  Google Scholar 

  126. Taylor R. P., Sci. Am., 287(6) (2002) 84.

    Article  Google Scholar 

  127. Pathria R. K., Am. J. Phys., 66 (1998) 1080.

    Article  ADS  Google Scholar 

  128. Beck G., Physica A, 306 (2002) 189.

    Article  ADS  MathSciNet  Google Scholar 

  129. Beck G., Physica A, 286 (2000) 164.

    Article  ADS  Google Scholar 

  130. Silva A. A. P., Vasconcellos A. R. and Luzzi R., unpublished.

  131. Rosenau P., Phys. Rev. E, 48 (1993) R655.

    Article  ADS  Google Scholar 

  132. Hamburg D. A. (Editor), Science, Technology and Govenment for a Changing World, Report of the Carnegie Commission, New York, USA (1993).

    Google Scholar 

  133. Golden W. T. and Lederberg J. (Editors), Enabling the Future: Linking Science and Technology to Societal Goals, Report of the Carnegie Commission, New York, USA (1994).

    Google Scholar 

  134. Gradshteyn I. S. and Ryzhik I. M., Table of Integrals, Series and Products (Academic Press, New York, USA) 1965.

    MATH  Google Scholar 

  135. Ramos J. G., Vasconcellos A. R. and Luzzi R., J. Chem. Phys., 112 (2000) 2692.

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute de Física “Gleb Wataghin”, Universidade Estadual de Campinas, Unicamp 13083-970 Campinas, São Paulo, Brazil

    R. Luzzi, Á. R. Vasconoellos & J. G. Ramos

Authors
  1. R. Luzzi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Á. R. Vasconoellos
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. J. G. Ramos
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Luzzi, R., Vasconoellos, Á.R. & Ramos, J.G. Non-equilibrium statistical mechanics of complex systems: An overview. Riv. Nuovo Cim. 30, 95–157 (2007). https://doi.org/10.1393/ncr/i2007-10018-6

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1393/ncr/i2007-10018-6

Key words

Search

Navigation