On the Exact Solution of a Generalized Polya Process

There are two types of master equations in describing nonequilibrium phenomena with memory effect: (i) the memory function type and (ii) the nonstationary type. A generalized Polya process is studied within the framework of a non-stationary type master equation approach. For a transition-rate with an arbitrary time-dependent relaxation function, the exact solution of a generalized Polya process is obtained. The characteristic features of temporal variation of the solution are displayed for some typical time-dependent relaxation functions reflecting memory in the systems. 1. Introduction The generalized master equation of memory function type [1] is a useful basis for analyzing non-equilibrium phenomena in open systems as where the kernel is conventionally assumed to have the product of a memory function with a transition rate as . The transition rate has the constraint with . This generalized master equation approach corresponds to the generalized Langevin equation of the memory function type [2, 3]. One can see many successful applications with long memory along the line of traditional formulation [1]. Looking around recent studies in complex open systems, there is an alternative approach based on a generalized non-stationary master equation [4] as The master equation in this form corresponds to the generalized Langevin equation of the convolutionless type, which is derived with the aid of projection operator method by Tokuyama and Mori [5]. The time-dependent coefficient may be written in the following form: . It is expected from the projection operator method [5] that the time-dependent function reflects the memory effect from varying environment in a different way associated with the memory function (cf. also H？nggi and Talkner [6]). The memory function (MF) formalism has been utilized in anomalous diffusion like Lévy type diffusion in atmospheric pollution, diffusion impurities in amorphous materials, and so on. The alternative convolution-less, non-stationary (NS) formalism gives only a small number of applications. The paper intends to exhibit a potential ability of the NS formalism by taking an arbitrary time-dependent function which is representing memory effect. The paper is organized as follows. Section 2 reviews the non-stationary Poisson process. Section 3 shows a generalized Polya process wherein it is involved a generalized non-stationary transition rate with an arbitrary function of time. The exact solution and the expression mean and variance are displayed as a function of . Some important remarks are given for a generalized