Russian Journal of Mathematical Physics

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

Coarsening in ergodic theory

  • V. V. Kozlov


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.


Phase Space Hamiltonian System Invariant Measure Ergodic Theory Liouville Equation 
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© Pleiades Publishing, Ltd. 2015

Authors and Affiliations

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

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