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.
All Science Journal Classification (ASJC) codes
- Statistical and Nonlinear Physics
- Mathematical Physics
- Physics and Astronomy(all)
- Applied Mathematics