Abstract
The master equation is rarely exactly solvable and hence various means of approximation have been devised. A popular systematic approximation method is the system-size expansion which approximates the master equation by a generalised Fokker–Planck equation. Here we first review the use of the expansion by applying it to a simple chemical system. The example shows that the solution of the generalised Fokker–Planck equation obtained from the expansion is generally not positive definite and hence cannot be interpreted as a probability density function. Based on this observation, one may also a priori conclude that moments calculated from the solution of the generalised Fokker–Planck equation are not accurate; however calculation shows these moments to be in good agreement with those obtained from the exact solution of the master equation. We present an alternative simpler derivation which directly leads to the same moments as the system-size expansion but which bypasses the use of generalised Fokker–Planck equations, thus circumventing the problem with the probabilistic interpretation of the solution of these equations.
Highlights
The Markovian description of stochastic systems with discrete state space is generally described by means of a master equation [1]
In the limit of infinitely large system sizes, consideration of the leading order terms of the system-size expansion (SSE) shows that the mean concentrations of the master equation agree with those of the deterministic rate equations, while the distribution of fluctuations about the mean is Gaussian and given by a Fokker-Planck equation with linear drift and diffusion coefficients
These higher-order terms have recently been used [7,8,9,10,11] to compute corrections to the mean concentration solution of the rate equations and to the second moments given by the linearnoise approximation” (LNA). These corrections are applicable when the system size is of intermediate size. They are computationally advantageous and typically highly accurate when compared to stochastic simulations [13,14,15], the generalised Fokker-Planck equations (GFPE) from which they are computed has been subject to strong criticisms
Summary
The Markovian description of stochastic systems with discrete state space is generally described by means of a master equation [1]. In the limit of infinitely large system sizes, consideration of the leading order terms of the SSE shows that the mean concentrations of the master equation agree with those of the deterministic rate equations, while the distribution of fluctuations about the mean is Gaussian and given by a Fokker-Planck equation with linear drift and diffusion coefficients The latter is often referred to as the “linearnoise approximation” (LNA) and is widely used in the literature (see for example [1, 3,4,5]). Consideration of higher orders of the expansion lead to generalised Fokker-Planck equations (GFPE) with third or higher order derivatives [6] These higher-order terms have recently been used [7,8,9,10,11] to compute corrections to the mean concentration solution of the rate equations and to the second moments given by the LNA (for a review of these results see [12]).
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