Several Study Of The Dynamical Behavior Of The Urn Model

We introduce urn models to describe the granular cluster, diffusion and phase transitions in shaken sand, and study the dynamical behaviors of these models.Firstly granular distributions in many-urn model is discussed under some dynamical rule. In the course of the system getting stable, we get different granular distribution at different time. A new parameter (?) is introduced to study the localizedbehaviors of particle distribution. Then the uniform, partly localized and localized steady states are also discussed.Secondly 2-urn model which can simulate the classical experiment concerns to the spatial separation of sand is extended. We consider that one or two balls in a urn can be chosen and jump into the urn beside it. With master equation, we get the probability distribution of the granular occupation; affirm the existence of the steady symmetric, steady asymmetric and metastable phases. Susceptibility k also is studied to prove that discontinuous transition is existed.In the end, we study 2-urn model and dynamics action, with the method of studying random walk. Another new parameter dx is introduced and discussed, it provides a new way to discuss system phase transition.