Advertisement

Russian Journal of Mathematical Physics

, Volume 22, Issue 2, pp 184–187 | Cite as

Coarsening in ergodic theory

  • V. V. Kozlov
Article
  • 31 Downloads

Abstract

This paper deals with the coarsening operation in dynamical systems where the phase space with a finite invariant measure is partitioned into measurable pieces and the summable function transferred by the phase flow is averaged over these pieces at each instant of time. Letting the time tend to infinity and then refining the partition, we arrive at a modernization of the von Neumann ergodic theorem, which is useful for the purposes of nonequilibrium statistical mechanics. In particular, for fine-grained partitions, we obtain the law of increment of coarse entropy for systems approaching the state of statistical equilibrium.

Keywords

Phase Space Hamiltonian System Invariant Measure Ergodic Theory Liouville Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    V. V. Nemytskii and V. V. Stepanov, Qualitative Theory of Differential Equations (Princeton,N.J.: Princeton Univ. Press, 1960; see also: New York: Dover, 1989).zbMATHGoogle Scholar
  2. 2.
    V. V. Kozlov and D. V. Treschev, “Fine-Grained and Coarse-Grained Entropy in Problems of Statistical Mechanics,” Theoret. and Math. Phys. 151(1), 539–555 (2007) [Teoret. Mat. Fiz. 151 (1), 120–137 (2007)].MathSciNetADSCrossRefzbMATHGoogle Scholar
  3. 3.
    W. Gibbs, Elementary Principles in Statistical Mechanics, Developed with Especial Reference to the Rational Foundation of Thermodynamics (New York: Dover, 1960).zbMATHGoogle Scholar
  4. 4.
    V. V. Kozlov, “Kinetics of Collisionless Continuous Medium,” Regul. Chaotic Dyn. 6(3), 235–251 (2001).MathSciNetADSCrossRefzbMATHGoogle Scholar
  5. 5.
    G. Piftankin and D. Treschev, “Coarse-Grained Entropy in Dynamical Systems,” Regul. Chaotic Dyn. 15(4–5), 575–597 (2010).MathSciNetADSCrossRefzbMATHGoogle Scholar
  6. 6.
    V. V. Kozlov, “The Generalized Vlasov Kinetic Equation,” Usp. Mat. Nauk 63(4), 93–130 (2008) [Russian Math. Surveys 63 (4), 691–726 (2008)].CrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2015

Authors and Affiliations

  1. 1.V.A. Steklov Mathematical Institute of the Russian Academy of SciencesMoscowRussia

Personalised recommendations