Maxwell’s demon as a dynamical model

We consider a model of a billiard-type system, which consists of two chambers connected through a hole. One chamber has a circle-shaped scatterer inside (Sinai billiard with infinite horizon), and the other one has a Cassini oval with a concave border. The phase space of the Cassini billiard contains islands, and its parameters are taken in such a way as to produce a self-similar island hierarchy. Poincaré recurrences to the left and to the right chambers are considered. It is shown that the corresponding distribution function does not reach “equipartition” even during the time ${10}^{10}$. The explanation is based on the existence of singularities in the phase space, which induces anomalous kinetics. The analogy to the Maxwell’s Demon model is discussed.