Journal of Mathematical Sciences

, Volume 128, Issue 2, pp 2791–2797 | Cite as

Weak Convergence of Measures in Conservative Systems

  • V. V. Kozlov
  • D. V. Treschev


Families of probability measures on the phase space of a dynamical system are considered. These measures are obtained as shifts of a given measure by the phase flow. Sufficient conditions for the existence of the weak convergence of the measures as the rate of the shift tends to infinity are suggested. The existence of such a limit leads to a new interpretation of the second law of thermodynamics. Bibliography: 5 titles.


Dynamical System Phase Space Probability Measure Weak Convergence Phase Flow 
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  1. 1.
    W. Gibbs, Elementary Principles in Statistical Mechanics Developed with Especial Reference to the Rational Foundation of Thermodynamics, New York (1902).Google Scholar
  2. 2.
    H. Poincare, “Reflections sur la theorie cinetique des gaz,” J. Phys. Theoret. et Appl., 4e Ser., 5, 369–403 (1906).Google Scholar
  3. 3.
    V. V. Kozlov, “Heat equilibrium according to Gibbs and Poincare,” Dokl. Akad. Nauk, 382, No.5, 602–606 (2002).MathSciNetGoogle Scholar
  4. 4.
    V. V. Nemytsky and V. V. Stepanov, Qualitative Theory of Differential Equations [in Russian], Moscow (1947).Google Scholar
  5. 5.
    V. V. Kozlov, “Kinetic of collisionless continuous medium,” Regul. Chaotic Dyn., 6, No.3, 235–251 (2001).CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • V. V. Kozlov
    • 1
  • D. V. Treschev
    • 1
  1. 1.Moscow State UniversityMoscowRussia

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