Non-equilibrium entropy on stationary Markov processes

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 . We prove that the convergence of the non-equilibrium entropy to zero for all measures of finite entropy is independent of for a wide class of convex functions, including ₀(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 , which include the L ^p -spaces for p>1. By means of the quadratic function ₂(t)=t ²–1, we relate the non-equlibrium entropies defined by the past -algebras of a K-dynamical system with the non-equilibrium entropy of its associated irreversible Markov processes converging to equilibrium.Partially supported by DIB Universidad de Chile, E19468412.