An alternative route to the system-size expansion

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.