Non-equilibrium entropy on stationary Markov processes

Abstract

We study the evolution of probability measures under the action of stationary Markov processes by means of a non-equilibrium entropy defined in terms of a convex function ϕ. We prove that the convergence of the non-equilibrium entropy to zero for all measures of finite entropy is independent of ϕ for a wide class of convex functions, including ϕ0(t)=t log t. We also prove that this is equivalent to the convergence of all the densities of a finite norm to a uniform density, on the Orlicz spaces related to ϕ, which include the L p -spaces for p>1. By means of the quadratic function ϕ2(t)=t 2−1, we relate the non-equlibrium entropies defined by the past σ-algebras of a K-dynamical system with the non-equilibrium entropy of its associated irreversible Markov processes converging to equilibrium.