Journal of Statistical Physics

, Volume 126, Issue 1, pp 117–132 | Cite as

Nonequilibrium Gas and Generalized Billiards



Generalized billiards describe nonequilibrium gas, consisting of finitely many particles, that move in a container, whose walls heat up or cool down. Generalized billiards can be considered both in the framework of the Newtonian mechanics and of the relativity theory. In the Newtonian case, a generalized billiard may possess an invariant measure; the Gibbs entropy with respect to this measure is constant. On the contrary, generalized relativistic billiards are always dissipative,and the Gibbs entropy with respect to the same measure grows under some natural conditions.

In this article, we find the necessary and sufficient conditions for a generalized Newtonian billiard to possess a smooth invariant measure, which is independent of the boundary action: the corresponding classical billiard should have an additional first integral of special type. In particular,the generalized Sinai billiards do not possess a smooth invariant measure. We then consider generalized billiards inside a ball, which is one of the main examples of the Newtonian generalized billiards which does have an invariant measure. We construct explicitly the invariant measure, and find the conditions for the Gibbs entropy growth for the corresponding relativistic billiard both formonotone and periodic action of the boundary.


nonequilibrium gas The Gibbs entropy invariant measure generalized billiards 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    H. Poincaré, Réflexions sur la théorie cinétique des gaz. J. Phys. Theoret. et Appl. 5(4):349–403 (1906).Google Scholar
  2. 2.
    Ya. G. Sinai, Dynamical systems with elastic reflections. Ergodic properties of dispersing billiards. Russ. Math. Surv. 25(2):137–189 (1970).CrossRefMathSciNetGoogle Scholar
  3. 3.
    G. Birkhoff. Dynamical systems. New York, AMS (1927).MATHGoogle Scholar
  4. 4.
    L. D. Pustyl'nikov, Poincaré models, rigorous justification of the second law of thermodynamics from mechanics, and the Fermi acceleration mechanism. Russian Math. Surveys 50(1):145–189 (1995).MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    L. D. Pustyl'nikov. The law of entropy increase and generalized billiards. Russian Math. Surv. 54(3):650–651 (1999).MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    L. D. Pustyl'nikov, A new mechanism for particle acceleration and a relativistic analogue of the Fermi–Ulam model. Theoret. and Math. Phys. 77(1):1110–1115 (1988).CrossRefMathSciNetGoogle Scholar
  7. 7.
    M. V. Deryabin and L. D. Pustyl'nikov, Generalized relativistic billiards. Reg. and Chaotic Dyn. 8(3):283–296 (2003).MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    M. V. Deryabin and L. D. Pustyl'nikov, On generalized relativistic billiards in external force fields. Lett. Math. Phys. 63(3):195–207 (2003).MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    M. V. Deryabin and L. D. Pustyl'nikov, Exponential attractors in generalized relativistic billiards. Comm. Math. Phys. 248(3):527–552 (2004).CrossRefADSMathSciNetGoogle Scholar
  10. 10.
    E. Fermi, On the origin of the cosmic radiation. Phys. Rev. 75:1169 (1949).MATHCrossRefADSGoogle Scholar
  11. 11.
    G. E. Uhlenbeck and G. W. Ford. Lectures in Statistical Mechanics. AMS, Providence, RI (1963).MATHGoogle Scholar
  12. 12.
    V. V. Kozlov and D. V. Billiards, Treshchev. A Genetic Introduction to the Dynamics of Systems with Impacts. Translations of Mathematical Monographs, 89. AMS, Providence, RI, 1991. viii+171 pp.MATHGoogle Scholar
  13. 13.
    J. W. Gibbs, Elementary principles in statistical mechanics. In: The Collected Works of J. W. Gibbs, Vol. II, Part 1, Yale University Press, New Haven, Conn. (1931).Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2006

Authors and Affiliations

  1. 1.The Mads Clausen InstituteUniversity of Southern DenmarkSønderborgDenmark
  2. 2.Keldysh Institute of Applied Mathematics of RASMoscowRussia
  3. 3.University of Bielefeld, BiBoSBielefeldGermany

Personalised recommendations