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 ${\mathbb{R}}^n$, 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, $t\in \mathbb{Z}$, $T_t=T^t$, or continuous, $t\in \mathbb{R}$. 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 ${\mathbb{R}}^n$, 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 $dx$.

7 pages, plain TeX; [email protected], [email protected], [email protected], to appear in Chaos: An Interdisciplinary Journal of Nonlinear Science, Volume 8, Issue 2

References in corpus (3)

• Steady-State Electrical Conduction in the Periodic Lorentz Gas
• Topics in chaotic dynamics
• Microscopic Reversibility and Macroscopic Behavior: Physical Explanations and Mathematical Derivations

Cited by in corpus (3)

• Gibbs entropy and irreversible thermodynamics
• Deterministic Diffusion in Periodic Billiard Models
• Anosov Properties of a Symplectic Map with Time-Reversal Symmetry