Approximate time-dependent solution of a master equation with full linear birth-death rates

Abstract

There are growing interests on dynamics of phase-singularities (PSs) in complex systems such as ventricular fibrillation, defect in fluids and liquid crystals, living creatures, quantum vortex and so on. A master equation approach on the number of PS for studying birth-death dynamics of PSs is invented first by Gil, Lega and Meunier. Although their approach is applied to various complex systems including non-linear birth-death rates, time-dependent solution of related master equation is obtained only rarely. Even a master equation with full linear birth-death rates, time-dependent solution is not also given due to the analytical complexity and the existence of singularity in the probability generating function. In this paper, an approximate time-dependent solution of the master equation and the associated waiting time distribution are obtained explicitly with the aid of the method of the Poisson transform. Numerical evaluation of the obtained approximate solution teaches us that there exists the universal scaling law in the waiting time distribution.

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“…Unfortunately finding an analytical solution for this equation becomes most often quite difficult, if not even impossible; and even though suitable approximations improved its overall application to this day, numerical analysis and/or stochastic simulations were usually shown to offer a better prospect of success. [200][201][202] In doing so, the common way to apply a numerical analysis to a probability equation is typically defined by consecutive random sampling of the system. 200,203 By generating a (pseudo)random number that is affiliated with one (or more) stochastic variables of the system, an associated algorithm normally simulates one possible trajectory of the underlying differential equation and consequently decides when or if an interaction, i.e., a change of states, takes place (Monte Carlo simulation).…”