A nonequilibrium entropy for dynamical systems

S. Goldstein, O. Penrose

Research output: Contribution to journalArticlepeer-review

38 Scopus citations


It is proposed to define entropy for nonequilibrium ensembles using a method of coarse graining which partitions phase space into sets which typically have zero measure. These are chosen by considering the totality of future possibilities for observation on the system. It is shown that this entropy is necessarily a nondecreasing function of the time t. There is no contradiction with the reversibility of the laws of motion because this method of coarse graining is asymmetric under time reversal. Under suitable conditions (which are stated explicitly) this entropy approaches the equilibrium entropy as t→+∞ and the fine-grained entropy as t→-∞. In particular, the conditions can always be satisfied if the system is a K-system, as in the Sinai billiard models. Some theorems are given which give information about whether it is possible to generate the partition used here for coarse graining from time translates of a finite partition, and at the same time elucidate the connection between our concept of entropy and the entropy invariant of Kolmogorov and Sinai.

Original languageEnglish (US)
Pages (from-to)325-343
Number of pages19
JournalJournal of Statistical Physics
Issue number2
StatePublished - Feb 1981

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics


  • Entropy
  • H-theorem
  • K-system
  • dynamical system
  • irreversibility

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