  We studied the Ehrenfest urn model in which particles in the same urn
interact with each other. Depending on the nature of interaction, the system
undergoes a first-order or second-order phase transition. The relaxation time
to the equilibrium state, the Poincare cycles of the equilibrium state and the
most far-from-equilibrium state, and the duration time of the states during
first-order phase transition are calculated. It was shown that the scaling
behavior of the Poincare cycles could be served as an indication to the nature
of phase transition, and the ratio of duration time of the states could be a
strong evidence of the metastability during first-order phase transition.
