On the Gaussian approximation for master equations

TL;DR

This paper evaluates the Gaussian approximation for master equations, demonstrating its accuracy and potential advantages over van Kampen's method, especially in systems with few particles.

Contribution

It provides a theoretical justification and comparative analysis showing the Gaussian approximation's improved accuracy for stochastic processes described by master equations.

Findings

Gaussian approximation accurately estimates first and second moments.

It outperforms van Kampen's method in certain regimes.

The approximation is particularly useful for small particle systems.

Abstract

We analyze the Gaussian approximation as a method to obtain the first and second moments of a stochastic process described by a master equation. We justify the use of this approximation with ideas coming from van Kampen's expansion approach (the fact that the probability distribution is Gaussian at first order). We analyze the scaling of the error with a large parameter of the system and compare it with van Kampen's method. Our theoretical analysis and the study of several examples shows that the Gaussian approximation turns out to be more accurate. This could be specially important for problems involving stochastic processes in systems with a small number of particles.