Some comments on approximations to the master equation

A common strategy for approximating a master equation is to replace it by a diffusion-like equation. Many methods for deriving the form of such an equation have been suggested in the literature. We compare two of these in the light of an example in which the master equation can be solved exactly. One of these is the van KampenΩ-expansion, which generally does not give a useful approximation to the equilibrium solution, and the second is a technique which preserves the noise-free dynamics and gives the correct equilibrium solution. It is shown that the second moment calculated in the latter approximation is not an accurate one at short times. The difficulty is the restriction of the approximating equation to the diffusion form.