Irreversible weak limits of classical dynamical systems

Abstract

A general discussion is given of weak limits of classical dynamical systems depending on a parameter. The resulting maps are shown to be invertible if and only if they define a group of measure preserving point transformations. The irreversible case automatically leads to positive bistochastic maps and is characterized in terms of convergence properties of the corresponding automorphisms of the observable algebra. Necessary and sufficient conditions are given for the limit to define a time-independent Markov process. Two models are discussed, for a particle in a periodic potential, and for a particle interacting with fixed configurations of external obstacles.