Free Energy of Disordered Urn Models in the Canonical Ensemble

Abstract

Urn models have been studied a lot because they are analytically tractable and contain rich physical phenomena, e.g., slow relaxation or condensation phenomena. It has been revealed that the urn models are related to zero range processes and asymmetric simple exclusion processes which are stochastic processes for studying nonequilibrium statistical physics. Furthermore, the urn models have been used in research fields of complex networks. Recently, the analytical treatment for disordered versions of the urn models have been developed. Leuzzi and Ritort have analyzed a disordered urn model (disordered backgammon model) in the scheme of the grand canonical ensemble. On the other hand, in the scheme of the canonical ensemble, the replica method has been used in order to calculate the occupation distribution of the preferential urn model. The replica method is a powerful tool for random systems, but still has an ambiguity for the mathematical validation. Furthermore, the analysis in ref. 9 has been based on replica symmetric solutions, so that it was not clear that the replica symmetric solution is adequate for the disordered urn model, although the solutions are in good agreement with numerical experiments. In this short note, we calculate the free energy of the disordered urn model using the law of large numbers. It is revealed that the saddle point equation obtained by the usage of the law of large numbers is the same as that obtained by the replica method in ref. 9. Hence, we conclude that the replica symmetric solution is adequate for the disordered urn model. Furthermore, we point out the mathematical similarity of free energies between the urn models and the random field Ising model (RFIM); this similarity gives an evidence that the replica symmetric solution of the urn models is exact. Firstly, we give a brief explanation of a general urn model (Fig. 1). An urn model consists of many urns and balls. We consider a system of M balls distributed among N urns; the number of balls contained in urn i is denoted by ni, and hence PN i1⁄41 ni 1⁄4 M. The density of the system is defined by 1⁄4 M=N. There are two types of urn models: the Ehrenfest class and the Monkey class. In the Ehrenfest class, balls within an urn are distinguishable. In contrast, the Monkey class has indistinguishable balls. These two types of urn models are treated in the similar analytical method, so that we discuss here only the case of the Ehrenfest class. The dynamics of the Ehrenfest urn model is briefly summarized as follows: (1) Choose a ball at random. (2) Select an urn with a transition probability ui. (3) Transfer the chosen ball to the selected urn i. (4) Repeat the above procedures until the system reaches an equilibrium state. Selecting an arbitrary transition probability ui, one can construct various urn models suitable for own physical problems. The total energy of the whole system is defined as