Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Journal of Non-Equilibrium Thermodynamics

Founded by Keller, Jürgen U.

Editor-in-Chief: Hoffmann, Karl Heinz

Managing Editor: Prehl, Janett / Schwalbe, Karsten

Ed. by Michaelides, Efstathios E. / Rubi, J. Miguel

4 Issues per year

IMPACT FACTOR 2017: 1.633
5-year IMPACT FACTOR: 1.642

CiteScore 2017: 1.70

SCImago Journal Rank (SJR) 2017: 0.591
Source Normalized Impact per Paper (SNIP) 2017: 1.160

See all formats and pricing
More options …
Volume 43, Issue 2


Ergodicity, Maximum Entropy Production, and Steepest Entropy Ascent in the Proofs of Onsager’s Reciprocal Relations

Francesco BenfenatiORCID iD: http://orcid.org/0000-0002-8224-3190 / Gian Paolo Beretta
  • 9297 Università degli Studi di Brescia, Department of Mechanical and Industrial Engineering, via Branze 38, Brescia, Italy
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2018-03-31 | DOI: https://doi.org/10.1515/jnet-2017-0054


We show that to prove the Onsager relations using the microscopic time reversibility one necessarily has to make an ergodic hypothesis, or a hypothesis closely linked to that. This is true in all the proofs of the Onsager relations in the literature: from the original proof by Onsager, to more advanced proofs in the context of linear response theory and the theory of Markov processes, to the proof in the context of the kinetic theory of gases. The only three proofs that do not require any kind of ergodic hypothesis are based on additional hypotheses on the macroscopic evolution: Ziegler’s maximum entropy production principle (MEPP), the principle of time reversal invariance of the entropy production, or the steepest entropy ascent principle (SEAP).

Keywords: Onsager relations; ergodicity; maximum entropy production; MEPP; steepest entropy ascent


  • [1]

    L. Onsager, Reciprocal relations in irreversible processes. I, Phys. Rev. 37 (1931), 405–426.CrossrefGoogle Scholar

  • [2]

    L. Onsager, Reciprocal relations in irreversible processes. II, Phys. Rev. 38 (1931), 2265–2279.CrossrefGoogle Scholar

  • [3]

    E.P. Wigner, Derivations of Onsager’s Reciprocal Relations, J. Chem. Phys. 22 (1954), 1912–1915.CrossrefGoogle Scholar

  • [4]

    S.R. de Groot and P. Mazur, On the statistical basis of Onsager’s reciprocal relations, Physica 23 (1957), 73–81.CrossrefGoogle Scholar

  • [5]

    M. Pavelka, V. Klika, and M. Grmela, Time reversal in nonequilibrium thermodynamics, Phys. Rev. E 90 (2014), 062131.Google Scholar

  • [6]

    J. Meixner, Consistency of the Onsager-Casimir reciprocal relations, Adv. Molec. Relax. Proc. 5 (1973), 319–331.CrossrefGoogle Scholar

  • [7]

    M. Liu, The Onsager symmetry relation and the time inversion invariance of the entropy production, preprint (1998), arXiv:cond-mat/9806318.Google Scholar

  • [8]

    H. Ziegler, An attempt to generalize Onsager’s principle, and its significance for rheological problems, Z. Angew. Math. Phys. 9 (1958), 748–763.CrossrefGoogle Scholar

  • [9]

    H. Ziegler, Proof of an orthogonality principle in irreversible thermodynamics, Z. Angew. Math. Phys. 21 (1970), 853–863.CrossrefGoogle Scholar

  • [10]

    H. Ziegler, An introduction to thermomechanics, Appl. Math. Mech. Series 21, North Holland, Amsterdam, 1977.Google Scholar

  • [11]

    M. Polettini, Fact-checking Ziegler’s maximum entropy production principle beyond the linear regime and towards steady states, Entropy 15 (2013), 2570–2584.CrossrefGoogle Scholar

  • [12]

    B.D. Coleman and C. Truesdell, On the Reciprocal Relations of Onsager, J. Chem. Phys. 33 (1960), 28–31.CrossrefGoogle Scholar

  • [13]

    E.P. Gyftopoulos and G.P. Beretta, What is a simple system?, J. Energy Res. Technol. 137 (2015), 021007.

  • [14]

    E.P. Gyftopoulos and G.P. Beretta, Thermodynamics: Foundations and Applications, Dover Publications, Mineaola, 2005.Google Scholar

  • [15]

    E. Zanchini and G.P. Beretta, Recent progress in the definition of thermodynamic entropy, Entropy 16 (2014), 1547–1570.CrossrefGoogle Scholar

  • [16]

    A. Carati, A. Maiocchi, L. Galgani, Statistical thermodynamics for metaequilibrium or metastable states, Meccanica 52 (2016), 1295–1307.Google Scholar

  • [17]

    H.B. Callen, Thermodynamics and an Introduction to Thermostatistics, Wiley, 1985.Google Scholar

  • [18]

    L. Onsager and S. Machlup, Fluctuations and irreversible processes, Phys. Rev. 91 (1953), 1505–1512.CrossrefGoogle Scholar

  • [19]

    C. Kittel, On the nonexistence of temperature fluctuations in small systems, Am. J. Phys. 41 (1973), 1211–1212.CrossrefGoogle Scholar

  • [20]

    C. Kittel, Temperature fluctuation: An oxymoron, Physics Today 41 (1988), 93.CrossrefGoogle Scholar

  • [21]

    H.B.G. Casimir, On Onsager’s principle of microscopic reversibility, Rev. Mod. Phys. 17 (1945), 343.CrossrefGoogle Scholar

  • [22]

    S.N. Patitsas, Onsager symmetry relations and ideal gas effusion: A detailed example, Am. J. Phys. 82 (2014), 123–134.CrossrefGoogle Scholar

  • [23]

    B.R. La Cour and W.C. Schieve, Derivation of the Onsager principle from large deviation theory, Physica A 331 (2004), 109–124.CrossrefGoogle Scholar

  • [24]

    U. Geigenmüller, U.M. Titulaer, and B.U. Felderhof, The approximate nature of the Onsager-Casimir reciprocal relations, Physica A 119 (1983), 53–66.CrossrefGoogle Scholar

  • [25]

    A.N. Gorban, I.V. Karlin, and A.Yu. Zinovyev, Constructive methods of invariant manifolds for kinetic problems, Phys. Reports 396 (2004), 197–403.CrossrefGoogle Scholar

  • [26]

    G.P. Beretta and E.P. Gyftopoulos, Thermodynamic derivations of conditions for chemical equilibrium and of Onsager reciprocal relations for chemical reactors, J. Chem. Phys. 121 (2004), 2718–2728.CrossrefGoogle Scholar

  • [27]

    M. Grmela, Contact Geometry of Mesoscopic Thermodynamics and Dynamics, Entropy 16 (2014), 1652–1686.CrossrefGoogle Scholar

  • [28]

    S.R. de Groot and P. Mazur, Non-equilibrium Thermodynamics, Dover publications, New York, 1984.Google Scholar

  • [29]

    A.E. Allahverdyan, Th.M. Nieuwenhuizen, Steady adiabatic state: its thermodynamics, entropy production, energy dissipation, and violation of Onsager relations, Phys. Rev. E 62 (2000), 845–850.CrossrefGoogle Scholar

  • [30]

    E.T. Jaynes, The minimum entropy production principle, Ann. Rev. Phys. Chem. 31 (1980), 579–601.CrossrefGoogle Scholar

  • [31]

    L.D. Landau and E.M. Lifshitz, Course of Theoretical Physics, Vol. 5: Statistical Physics, Elsevier Science, 2013.Google Scholar

  • [32]

    M. Campisi and D.H. Kobe, Derivation of the Boltzmann principle, Am. J. Phys. 78 (2010), 608–615.CrossrefGoogle Scholar

  • [33]

    U.M.B. Marconi, A. Puglisi, L. Rondoni, and A. Vulpiani, Fluctuation–dissipation: Response theory in statistical physics, Phys. Reports 461 (2008), 111–195.CrossrefGoogle Scholar

  • [34]

    M.S. Green, Markoff random processes and the statistical mechanics of time-dependent phenomena, J. Chem. Phys. 20 (1952), 1281–1295.CrossrefGoogle Scholar

  • [35]

    G. Gallavotti, Chaotic hypothesis: Onsager reciprocity and fluctuation-dissipation theorem, J. Stat. Phys. 84 (1996), 899–925.CrossrefGoogle Scholar

  • [36]

    G. Gallavotti, Extension of Onsager’s reciprocity to large fields and the chaotic hypothesis, Phys. Rev. Lett. 77 (1996), 4334.CrossrefGoogle Scholar

  • [37]

    G. Gallavotti, Thermostats, chaos and Onsager reciprocity, J. Stat. Phys. 134 (2009), 1121.CrossrefGoogle Scholar

  • [38]

    L.P. Pitaevskii and E.M. Lifshitz, Course of Theoretical Physics, Vol. 10: Physical Kinetics, Elsevier Science, 2013.Google Scholar

  • [39]

    F. Sharipov, Onsager-Casimir reciprocal relations based on the Boltzmann equation and gas-surface interaction: Single gas, Phys. Rev. E 73 (2006), 026110.Google Scholar

  • [40]

    L. Rosenfeld, Classical Statistical Mechanics (inglês), Editora Livraria da Fisica, 2005.Google Scholar

  • [41]

    C. Maes and K. Netočný, Time-reversal and entropy, J. Stat. Phys. 110 (2003), 269–310.CrossrefGoogle Scholar

  • [42]

    A. Montefusco, F. Consonni, and G.P. Beretta, Essential equivalence of the general equation for the nonequilibrium reversible-irreversible coupling (GENERIC) and steepest-entropy-ascent models of dissipation for nonequilibrium thermodynamics, Phys. Rev. E 91 (2015), 042138.Google Scholar

  • [43]

    L.M. Martyushev and V.D. Seleznev, Maximum entropy production: application to crystal growth and chemical kinetics, Current Opinion in Chemical Engineering 7 (2015), 23–31. See also references therein and also S. Gheorghiu-Svirschevski, Addendum to “Nonlinear quantum evolution with maximal entropy production”, Phys. Rev. A 63 (2001), 054102.CrossrefGoogle ScholarGoogle Scholar

  • [44]

    G.P. Beretta, Steepest entropy ascent in quantum thermodynamics, Lect. Notes Phys. 278 (1987), 441–443. See also G.P. Beretta, Nonlinear quantum evolution equations to model irreversible adiabatic relaxation with maximal entropy production and other nonunitary processes, Reps. Math. Phys. 64 (2009), 139–168.CrossrefGoogle ScholarGoogle Scholar

  • [45]

    G.P. Beretta, Steepest entropy ascent model for far-non-equilibrium thermodynamics: unified implementation of the maximum entropy production principle, Phys. Rev. E 90 (2014), 042113.Google Scholar

  • [46]

    G.P. Beretta, Quantum thermodynamics of nonequilibrium. Onsager reciprocity and dispersion-dissipation relations, Found. Phys. 17 (1987), 365–381.CrossrefGoogle Scholar

  • [47]

    C. Reina and J. Zimmer, Entropy production and the geometry of dissipative evolution equations, Phys. Rev. E 92 (2015), 052117.Google Scholar

  • [48]

    A. Mielke, M.A. Peletier, and D.R.M. Renger, A generalization of Onsager’s reciprocity relations to gradient flows with nonlinear mobility, J. Non-Equil. Therm. 41 (2016), 141–149.Google Scholar

  • [49]

    R. Jordan, D. Kinderlehrer, and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal. 29 (1998), 1–17.CrossrefGoogle Scholar

  • [50]

    L. Ambrosio, N. Gigli, and G. Savaré, Gradient flows: in metric spaces and in the space of probability measures, Birkhäuser, 2005.Google Scholar

  • [51]

    P. Ván and B. Nyıri, Hamilton formalism and variational principle construction, Annalen der Physik (Leipzig) 8 (1999), 331–354.CrossrefGoogle Scholar

About the article

Received: 2017-09-27

Revised: 2018-01-08

Accepted: 2018-02-02

Published Online: 2018-03-31

Published in Print: 2018-04-25

Citation Information: Journal of Non-Equilibrium Thermodynamics, Volume 43, Issue 2, Pages 101–110, ISSN (Online) 1437-4358, ISSN (Print) 0340-0204, DOI: https://doi.org/10.1515/jnet-2017-0054.

Export Citation

© 2018 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

Gerardo Valadez Huerta, Vincent Flasbart, Tobias Marquardt, Pablo Radici, and Stephan Kabelac
Entropy, 2018, Volume 20, Number 6, Page 469

Comments (0)

Please log in or register to comment.
Log in