EVOLUTION OF ENSEMBLES TOWARD AN ABSORBING ATTRACTOR

Abstract

In Ref. [21] the evolution of ensembles of states for dynamical systems (DS) With states in a probability space is studied. Enlarging some results in Ref. [21] to a very common measure spaces the above study is generalized for DSs with states in those measure spaces. This enlargement uncovers an indecomposability subregion (AA) of the space of states, for which each orbit of states of the system has all its states in AA from some time on. It is obtained that each ensemble of states, in its natural evolution, evolves in ergodic sense (convergence a la weak Cesaro) to the microcanonical ensemble (scattered by all AA). A particular but central kind of that evolution is the irreversible one in coarse-grained sense (weak convergence) to the microcanonical ensemble

it is obtained that such evolution holds if and only if the DS is mixing for measure spaces. For this kind of DS predictability [241 is impossible beyond a certain time, however the only forecasting in coarse-grained sense about the system's future which can be made is that of a statistical nature.