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 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 Z, T(t)=T(t), or continuous, t in 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 R(n), the expectation value of logrho(x,t), with respect to any stationary (i.e., time invariant) measure nu(dx), is linear in t, nu(logrho(x,t))=nu(logrho(x,0))+Kt. K depends only on nu and vanishes when nu is absolutely continuous with respect to dx.(c) 1998 American Institute of Physics.

Coarse-graining effects on time-dependent information on invariant sets

“…Goldstein et al [1] showed that the entropy functional m(log q) increases linearly in time. Herein m = m(dx) is the invariant measure with respect to the volume measure dx and q is the continuous probability density for the given system.…”

“…If the flow is expressed as an ordinary equation system dx/dt = v (v: smooth and differentiable), then the increasing rate K of the functional can be expressed as k = m(À$ AE v), i.e. phase space volume contraction rate [1].…”

“…doi:10.1016/j.chaos.2005.01.007 (dissipative) dynamical system when an entropy-production-like term is introduced. It can be regarded as the symbolic result of the connection that the entropy functional defined on the invariant set without any partitioning, proposed by Goldstein et al [1], has the increasing rate that agrees with the volume contraction rate.…”

“…In this paper the entropy functional proposed by Goldstein et al [1] was extended to a family of information with a parameter b, and the above-mentioned coarse graining effects were examined on dissipative systems by comparing timedependent characteristics of the information with those of coarse-grained information. The cause of the effects was also discussed in connection with mesoscopic irreversible processes.…”

“…Macroscopic irreversibility, especially entropy production, has been a central issue in non-equilibrium statistical mechanics [1][2][3][4][5][6]. In the Hamilton or volume-preserving system, the conventional entropy, Gibbs entropy, is held constant and therefore its desirable form that is consistent with the second law of thermodynamics has been investigated.…”