Irreversible weak limits of classical dynamical systems

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 transformation s. The irreversible case automatically leads to positive bistochastic maps and is characterized in terms of convergence properties of the correspondin g automorphisms of the observable algebra. Necessary and sufficient conditi ons are given for the limit to define a time-independent Markov process. Tw o models are discussed, for a particle in a periodic potential, and for a p article interacting with fixed configurations of external obstacles. (C) 19 99 American Institute of Physics. [S0022-2488(99)03208-9].