Remark on the (Non)convergence of Ensemble Densities in Dynamical Systems

Details Remark on the (Non)convergence of Ensemble Densities in Dynamical Systems Remark on the (Non)convergence of Ensemble Densities in Dynamical Systems

1998-04-27

arxiv.org/abs/math-ph/9804016v1

Physics

Mathematical Physics

7 pages, plain TeX; oldstein@math.rutgers.edu, lebowitz@math.rutgers.edu, sinai@math.princeton.edu, to appear in Chaos: An I

Scientific paper

10.1063/1.166321

We consider a dynamical system with state space $M$, a smooth, compact subset of some ${\Bbb R}^n$, and evolution given by $T_t$, $x_t = T_t x$, $x \in M$; $T_t$ is invertible and the time $t$ may be discrete, $t \in {\Bbb Z}$, $T_t = T^t$, or continuous, $t \in {\Bbb R}$. Here we show that starting with a continuous positive initial probability density $\rho(x,0) > 0$, with respect to $dx$, the smooth volume measure induced on $M$ by Lebesgue measure on ${\Bbb R}^n$, the expectation value of $\log \rho(x,t)$, with respect to any stationary (i.e. time invariant) measure $\nu(dx)$, is linear in $t$, $\nu(\log \rho(x,t)) = \nu(\log \rho(x,0)) + Kt$. $K$ depends only on $\nu$ and vanishes when $\nu$ is absolutely continuous wrt $dx$.

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