Statistical irreversibility of the Kac reversible circular model

The Kac circular model is a discrete dynamical system which has the property of recurrence and reversibility. Within the framework of this model M. Kac formulated necessary conditions for irreversibility over “short” time intervals to take place and demonstrated Boltzmann's most important exploration methods and ideas, outlining their advantages and limitations. We study the circular model within the realm of the theory of Gibbs ensembles and offer a new approach to a rigorous proof of the “zeroth” law of thermodynamics basing on the analysis of weak convergence of probability distributions.