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
“On the wonderful world of random walks,” in Nonequilibrium Phenomena II: From Stochastic to Hydrodynamics, E. W. Montroll and M. F. Shlesinger, Eds.J. L. Lebowitz and E. W. Montroll, Eds., chapter 1, pp. 1–121, North-Holland, Amsterdam, The Netherlands, 1984.
M. Tokuyama and H. Mori, “Statistical-mechanical theory of random frequency modulations and generalized Brownian motions,” Progress of Theoretical Physics, vol. 55, no. 2, pp. 411–429, 1976.
P. H？nggi and P. Talkner, “On the equivalence of time-convolutionless master equations and generalized Langevin equations,” Physics Letters A, vol. 68, no. 1, pp. 9–11, 1978.
Y. Ogata, “Statistical models for earthquake occurrences and residual analysis for point processes,” Journal of the American Statistical Association, vol. 83, pp. 9–27, 1988.
R. L. Smith, “Statistics of extremes, with applications in environment, insurance and finance, in extremevalue in finance,” in Telecommunications and the Environment, B. Finkenstadt and H. Rootzen, Eds., Chapman & Hall/CRC, London, UK, 2003.
H. Konno, “The stochastic process of non-linear random vibration. Reactor-noise analysis of hump phenomena in a time domain,” Annals of Nuclear Energy, vol. 13, no. 4, pp. 185–201, 1986.
K. H. W. J. ten Tusscher and A. V. Panfilov, “Alternans and spiral breakup in a human ventricular tissue model,” American Journal of Physiology, vol. 291, no. 3, pp. H1088–H1100, 2006.
O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases, Wiley Series in Mathematical and Computational Biology, John Wiley & Sons, Chichester, UK, 2000.
S. B. Lowen and M. C. Teich, “The periodogram and Allan variance reveal fractal exponents greater than unity in auditory-nerve spike trains,” Journal of the Acoustical Society of America, vol. 99, no. 6, pp. 3585–3591, 1996.