Poincaré cycle of a multibox Ehrenfest urn model with directed transport

A generalized Ehrenfest urn model of many urns arranged periodically along a circle was introduced. An N-ball, M-urn problem was solved explicitly. The evolution of the system was studied, and the average number of balls in a certain urn at any time step was calculated. It was shown that this mean value oscillates several times before it arrives the stationary value. The Poincare cycle was also obtained for two situations. The results indicate that the fundamental assumption of statistical mechanics holds in this system.