The European Physical Journal Special Topics

, Volume 226, Issue 10, pp 2327–2343 | Cite as

On the relevance of the maximum entropy principle in non-equilibrium statistical mechanics

Regular Article
Part of the following topical collections:
  1. Aspects of Statistical Mechanics and Dynamical Complexity

Abstract

At first glance, the maximum entropy principle (MEP) apparently allows us to derive, or justify in a simple way, fundamental results of equilibrium statistical mechanics. Because of this, a school of thought considers the MEP as a powerful and elegant way to make predictions in physics and other disciplines, rather than a useful technical tool like others in statistical physics. From this point of view the MEP appears as an alternative and more general predictive method than the traditional ones of statistical physics. Actually, careful inspection shows that such a success is due to a series of fortunate facts that characterize the physics of equilibrium systems, but which are absent in situations not described by Hamiltonian dynamics, or generically in nonequilibrium phenomena. Here we discuss several important examples in non equilibrium statistical mechanics, in which the MEP leads to incorrect predictions, proving that it does not have a predictive nature. We conclude that, in these paradigmatic examples, an approach that uses a detailed analysis of the relevant aspects of the dynamics cannot be avoided.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    E.T. Jaynes, Information theory and statistical mechanics, Phys. Rev. 106, 620 (1957)ADSMathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    E.T. Jaynes, Information theory and statistical mechanics II, Phys. Rev. 108, 171 (1957)ADSMathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    E.T. Jaynes, Information Theory and Statistical Mechanics, in K. Ford (ed.), Statistical Physics (Benjamin, New York, 1963), pp. 181–218Google Scholar
  4. 4.
    S. Chibbaro, L. Rondoni, A. Vulpiani, Reductionism, Emergence and Levels of Reality: The Importance of Being Borderline (Springer, Heidelberg, 2014)Google Scholar
  5. 5.
    S. Chibbaro, L. Rondoni, A. Vulpiani, On the foundations of statistical mechanics: ergodicity, many degrees of freedom and inference, Comm. Theor. Phys. 62, 469 (2014)CrossRefMATHGoogle Scholar
  6. 6.
    J. Uffink, Can the maximum entropy principle be explained as a consistency requirement?, Stud. History Philos. Mod. Phys. 26, 223 (1995)ADSMathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    J. Uffink, The constraint rule of the maximum entropy principle, Stud. History Philos. Mod. Phys. 27, 47 (1996)ADSMathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    A. Shimony, The Status of the Principle of Maximum Entropy, Synthese 63, 35. Search for a Naturalistic Point of View, Cambridge, University PressGoogle Scholar
  9. 9.
    K. Friedman, A. Shimny, Jaynes’s maximum entropy prescription and probability theory, J. Stat. Phys. 3, 381 (1971)ADSCrossRefGoogle Scholar
  10. 10.
    P.W. Atkins, J. De Paula, Physical Chemistry (Oxford, University Press, 2002, 2nd ed. 2006)Google Scholar
  11. 11.
    A.C. Elitzur, Locality and indeterminism preserve the second law, Phys. Lett. A167, 335 (1992)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    A.C. Elitzur, S. Dolev, Quantum Phenomena Within a New Theory of Time, in A.C., Elitzur, S. Dolev, N. Kolenda (eds.), Quo Vadis Quantum Mechanics? (Springer, Berlin, 2005) pp. 325–49Google Scholar
  13. 13.
    J.L. Lebowitz, Boltzmann’s entropy and time’s arrow, Phys. Today 46, 32 (1993)CrossRefGoogle Scholar
  14. 14.
    M. Falcioni, L. Palatella, S. Pigolotti, L. Rondoni, A. Vulpiani, Initial growth of Boltzmann entropy and chaos in a large assembly of weakly interacting systems, Phys. A 385, 170 (2007)MathSciNetCrossRefGoogle Scholar
  15. 15.
    O.E. Lanford, III, in: J. Moser (ed.), Dynamical Systems, Theory and Applications, Lecture Notes in Physics (Springer, Berlin, 1975), Vol. 38 Google Scholar
  16. 16.
    L. Cerino, F. Cecconi, M. Cencini, A. Vulpiani, The role of the number of degrees of freedom and chaos in macroscopic irreversibility, Physica A 442, 486 (2016)ADSCrossRefGoogle Scholar
  17. 17.
    A.L. Kuzemsky, Probability information and statistical physics, Int. J. Theor. Phys. 55, 1378 (2015)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    U. Frisch, Turbulence: The Legacy of A.N. Kolmogorov (Cambridge University Press, 1995)Google Scholar
  19. 19.
    T. Bohr, M.H. Jensen, G. Paladin, A. Vulpiani, Dynamical Systems Approach to Turbulence (Cambridge University Press, 1998)Google Scholar
  20. 20.
    P.D. Ditlevsen, I.A. Mogensen, Cascades and statistical equilibrium in shell models of turbulence, Phys. Rev. E 53, 4785 (1996)ADSCrossRefGoogle Scholar
  21. 21.
    M.H. Jensen, G. Paladin, A. Vulpiani, Intermittency in a cascade molel for 3-dimensional turbulence, Phys. Rev. A 43, 798 (1991)ADSCrossRefGoogle Scholar
  22. 22.
    R. Dewar, Information theory explanation of the fluctuation theorem maximum entropy production and self-organized criticality in non-equilibrium stationary states, J. Phys. A36, 631 (2003)ADSMathSciNetMATHGoogle Scholar
  23. 23.
    R. Dewar, Maximum entropy production and the fluctuation theorem, J. Phys. A38, L371 (2005)ADSMathSciNetMATHGoogle Scholar
  24. 24.
    T.M. Cover, J.A. Thomas, Elements of Information Theory (Wiley, New York, 1991)Google Scholar
  25. 25.
    L. Onsager, S. Machlup, Fluctuations and irreversible processes, Phys. Rev. 91, 1505 (1953)ADSMathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    S.K. Ma, Statistical Mechanics (World Scientific, Singapore, 1985)Google Scholar
  27. 27.
    D.J. Evans, D.J. Searles, L. Rondoni, Application of the Gallavotti-Cohen fluctuation relation to thermostated steady states near equilibrium, Phys. Rev. E 71, 056120 (2005)ADSMathSciNetCrossRefGoogle Scholar
  28. 28.
    L. Rondoni, C. Mejà-Monasterio, Fluctuations in nonequilibrium statistical mechanics: models, mathematical theory, physical mechanisms, Nonlinearity 20, R1 (2007)ADSCrossRefMATHGoogle Scholar
  29. 29.
    U. Marini Bettolo Marconi, A. Puglisi, L. Rondoni, A. Vulpiani, Fluctuation-dissipation: Response theory in statistical physics, Phys. Rep. 461, 111 (2008)ADSCrossRefGoogle Scholar
  30. 30.
    E.G.D. Cohen, G. Gallavotti, Note on two theorems in nonequilibrium statistical mechanics, J. Stat. Phys. 96, 1343 (1999)ADSMathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    D.J. Searles, L. Rondoni, D.J. Evans, The steady state fluctuation relation for the dissipation function, J. Stat. Phys. 128, 1337 (2007)ADSMathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    F. Bonetto, G. Gallavotti, A. Giuliani, F. Zamponi, Chaotic hypothesis, fluctuation theorem and singularities, J. Stat. Phys. 123, 39 (2006)ADSMathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    F. Bonetto, G. Gallavotti, Reversibility, coarse graining and the chaoticity principle, Commun. Math. Phys. 189, 263 (1997)ADSMathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    L. Rondoni, G.P. Morriss, Large fluctuations and axiom-Cstructures in deterministically thermostatted systems, Open Syst. Inf. Dynam. 10, 105 (2003)CrossRefMATHGoogle Scholar
  35. 35.
    G. Gallavotti, L. Rondoni, E. Segre, Lyapunov spectra and nonequilibrium ensembles equivalence in 2D fluid mechanics, Physica D 187, 338 (2004)ADSMathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    J. Farago, Injected power fluctuations in Langevin equation, J. Stat. Phys. 107, 781 (2002)ADSMathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    G.M. Wang, E.M. Sevick, E. Mittag, D.J. Searles, D.J. Evans, Experimental demonstration of violations of the second law of thermodynamics for small systems and short time scales, Phys. Rev. Lett. 89, 050601 (2002)ADSCrossRefGoogle Scholar
  38. 38.
    R. van Zon, E.G.D. Cohen, Stationary and transient work-fluctuation theorems for a dragged Brownian particle, Phys. Rev. E 67, 046102 (2003)ADSCrossRefGoogle Scholar
  39. 39.
    M. Baiesi, T. Jacobs, C. Maes, N.S. Skantzos, Fluctuation symmetries for work and heat, Phys. Rev. E 74, 021111 (2006)ADSCrossRefGoogle Scholar
  40. 40.
    T. Mai, A. Dhar, Nonequilibrium work fluctuations for oscillators in non-Markovian baths, Phys. Rev. E 75, 061101 (2007)ADSCrossRefGoogle Scholar
  41. 41.
    R.J. Harris, Ràkos, M. Schütz, Breakdown of Gallavotti-Cohen symmetry for stochastic dynamics, Europhys. Lett. 75, 227 (2006)ADSMathSciNetCrossRefGoogle Scholar
  42. 42.
    L. Conti, P. De Gregorio, G. Karapetyan, C. Lazzaro, M. Pegoraro, M. Bonaldi, L. Rondoni, Effects of breaking vibrational energy equipartition on measurements of temperature in macroscopic oscillators subject to heat flux, JSTAT 2013, P12003 (2013)CrossRefGoogle Scholar
  43. 43.
    G. Auletta, A Paradigm Shift in Biology? Information 1, 28 (2010)Google Scholar
  44. 44.
    G. Auletta, Cognitive Biology: Dealing with Information from Bacteria to Minds, (Oxford University Press, 2011)Google Scholar
  45. 45.
    I. Prigogine, Etude thermodynamique des phénomènes irréversibles (Desoer, Liège, 1947)Google Scholar
  46. 46.
    K.J. Friston, B. Sengupta, G. Auletta, Cognitive Dynamics: From Attractors to Active Inference, Proc. IEEE 102, 427 (2014)CrossRefGoogle Scholar
  47. 47.
    Ch. Feinauer, M.J. Skwark, A. Pagnani, E. Aurell, Improving Contact Prediction along Three Dimensions, PLOS Comp. Biol. 10, e1003847 (2014)ADSCrossRefGoogle Scholar

Copyright information

© EDP Sciences and Springer 2017

Authors and Affiliations

  1. 1.University of CassinoCassinoItaly
  2. 2.Dipartimento di Scienze Matematiche, Politecnico di TorinoTorinoItaly
  3. 3.INFN, Sezione di TorinoTorinoItaly
  4. 4.Kavli Institute for Theoretical Physics China, CASBeijingP.R. China
  5. 5.Università di Roma “Sapienza”, Dipartimento di FisicaRomeItaly
  6. 6.CNR, Istituto Sistemi ComplessiRomeItaly

Personalised recommendations