Approximating the master equation by Fokker–Planck-type equations for single-variable chemical systems

Abstract

The stochastic evolution of a well stirred chemically reacting system containing a single time-varying species X is accurately described by the master equation, in which the total number x of X molecules is an integer variable. We investigate here the legitimacy of approximating the master equation by a Fokker–Planck type partial differential equation in which x is treated as a real variable. Taking the position that any partial differential equation may be regarded as a legitimate approximation to the master equation if and only if it reduces to the master equation when subjected to a proper discretization procedure, we deduce the following: For the special case in which the various chemical reactions can alter the X molecule population by no more than one molecule at a time, a second order (two term) Fokker–Planck equation suffices. However, for the more general case in which reactions are allowed that alter the X molecule population by two molecules at a time, it is necessary to use at least a fourth order (four term) Fokker–Planck equation. In all cases it is assumed that x is large compared to unity; however, no assumptions are made about the way in which the transition probability rates and the molecular population fluctuations scale with the size of the system.