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  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 . Looking around recent studies in complex open systems, there is an alternative approach based on a generalized non-stationary master equation  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 . The time-dependent coefficient may be written in the following form: . It is expected from the projection operator method  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 ). 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
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