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

Abstract

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 $ρ(x,0) > 0$, with respect to $dx$, the smooth volume measure induced on $M$ by Lebesgue measure on ${\Bbb R}^n$, the expectation value of $\log ρ(x,t)$, with respect to any stationary (i.e. time invariant) measure $ν(dx)$, is linear in $t$, $ν(\log ρ(x,t)) = ν(\log ρ(x,0)) + Kt$. $K$ depends only on $ν$ and vanishes when $ν$ is absolutely continuous wrt $dx$.