Statistical irreversibility of the Kac reversible circular model



The Kac circular model is a discrete dynamical system which has the property of recurrence and reversibility. Within the framework of this model M. Kac formulated necessary conditions for irreversibility over “short” time intervals to take place and demonstrated Boltzmann’s most important exploration methods and ideas, outlining their advantages and limitations. We study the circular model within the realm of the theory of Gibbs ensembles and offer a new approach to a rigorous proof of the “zeroth” law of thermodynamics based on the analysis of weak convergence of probability distributions.


reversibility stochastic equilibrium weak convergence 

MSC2010 numbers



  1. 1.
    Ehrenfest, P. and Ehrenfest, T., Bemerkung zur Theorie der Entropiezunahme in der “Statistischen Mechanik” von W.Gibbs, Sitzungsberichte Akad. Wiss. Wien, 1906, vol. 115,pt. IIa, pp. 89–98.MATHGoogle Scholar
  2. 2.
    Ehrenfest, P. and Ehrenfest, T., Über zwei bekannte Eniwände gegen das Boltzmannsche H-Theorem, Phys. Zschr., Jg. 9, vol. 8, pp. 311–314.Google Scholar
  3. 3.
    Markov, A.A., Generalization of the Problem of the Consecutive Exchange of Balls, Izv. Akad. Nauk, 4 ser., 1918, vol. 12, pp. 261–266 (Russian).Google Scholar
  4. 4.
    Kac, M., Probability and Related Topics in Physical Sciences, New York-London: Intersci. Publ., 1958.Google Scholar
  5. 5.
    Kac, M., Some Stochastic Problems in Physics and Mathematics, Colloquium Lectures in Pure and Applied Science, Dallas, TX: Magnolia Petroleum Co., 1956.Google Scholar
  6. 6.
    Kozlov, V.V., Thermal Equilibrium in the Sense of Gibbs and Poincaré, Moscow-Izhevsk: Institute of Computer Science, 2002 (Russian).MATHGoogle Scholar
  7. 7.
    Kozlov, V.V., Gibbs Ensembles and Non-equilibrium Statistical Mechanics, Moscow-Izhevsk: Institute of Computer Science, 2008 (Russian).Google Scholar
  8. 8.
    Poincaré, H., Réfflexions sur la théorie cinétique des gaz, J. Phys. théoret. et appl., 4-e sér., 1906, vol. 5, pp. 369–403.Google Scholar
  9. 9.
    Gibbs, W., Elementary Principles in Statistical Mechanics, Developed with Especial Reference to the Rational Foundation of Thermodynamics, New York: Schribner, 1902.MATHGoogle Scholar
  10. 10.
    Hardy, G. H., Divergent series, Oxford: Oxford Univ. Press, 1949.MATHGoogle Scholar
  11. 11.
    Prigogine, I. and Stengers, I., Order out of Chaos: Man’s New Dialogue with Nature, London: Heinemann, 1984.Google Scholar
  12. 12.
    Kozlov, V.V. and Treshchev, D. V., Fine-Grained and Coarse-Grained Entropy in Problems of Statistical Mechanics, Teoret. Mat. Fiz., 2007, vol. 151, no. 1, pp. 120–137 [Theoret. and Math. Phys., 2007, vol. 151, no. 1, pp. 539–555].MathSciNetGoogle Scholar
  13. 13.
    Bogolyubov, N.N., Problems of Dynamical Theory in Statistical Physics, Moscow: Gostekhizdat, 1946 (Russian).Google Scholar
  14. 14.
    Uhlenbeck, G. E. and Ford, G.W., Lectures in Statistical Mechanics, Providence, RI: AMS, 1963.MATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2011

Authors and Affiliations

  1. 1.V.A. Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia

Personalised recommendations