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 \(\phi\). We prove that the convergence of the non-equilibrium entropy to zero for all measures of finite entropy is independent of \(\phi\) for a wide class of convex functions, including \(\phi_ 0(t)=t \log t.\) 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 \(\phi\), which include the \(L^ p\)-spaces for \(p>1\). By means of the quadratic function \(\phi_ 2(t)=t^ 2-1\), we relate the nonequilibrium entropies defined by the past \(\sigma\)-algebras of a K- dynamical system with the non-equilibrium entropy of its associated irreversible Markov processes converging to equilibrium.
scientific article; zbMATH DE number 3909470
scientific article