Remark on the (non)convergence of ensemble densities in dynamical systems

S. Goldstein, J. L. Lebowitz, Y. Sinai

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6 Scopus citations

Abstract

We consider a dynamical system with state space M, a smooth, compact subset of some ℝn, and evolution given by Tt, xt = Ttx, x ∈ M; Tt is invertible and the time t may be discrete, t ∈ ℤ, Tt = Tt, or continuous, t ∈ ℝ. Here we show that starting with a continuous positive initial probability density ρ(x,0) > 0, with respect to dx, the smooth volume measure induced on M by Lebesgue measure on ℝn, the expectation value of logρ(x,t), with respect to any stationary (i.e., time invariant) measure v(dx), is linear in t, v(logρ(x,t)) = v(logρ(x,0)) + Kt. K depends only on v and vanishes when v is absolutely continuous with respect to dx.

Original languageEnglish (US)
Pages (from-to)393-395
Number of pages3
JournalChaos
Volume8
Issue number2
DOIs
StatePublished - Jun 1998

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Physics and Astronomy(all)
  • Applied Mathematics

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