A field theoretic approach to master equations and a variational method beyond the Poisson ansatz

TL;DR

This paper introduces an advanced variational method within a field theoretic framework to solve complex master equations more accurately, surpassing the limitations of the Poisson ansatz, especially in modeling stochastic gene regulatory networks.

Contribution

It extends the variational approach to include arbitrary functions, improving the approximation of master equations beyond the Poisson ansatz.

Findings

Achieves good quantitative approximation for gene regulatory network master equations.

Reduces dimensionality of master equations significantly.

Enables treatment of fluctuation effects beyond Poisson distribution.

Abstract

We develop a variational scheme in a field theoretic approach to a stochastic process. While various stochastic processes can be expressed using master equations, in general it is difficult to solve the master equations exactly, and it is also hard to solve the master equations numerically because of the curse of dimensionality. The field theoretic approach has been used in order to study such complicated master equations, and the variational scheme achieves tremendous reduction in the dimensionality of master equations. For the variational method, only the Poisson ansatz has been used, in which one restricts the variational function to a Poisson distribution. Hence, one has dealt with only restricted fluctuation effects. We develop the variational method further, which enables us to treat an arbitrary variational function. It is shown that the variational scheme developed gives a quantitatively good approximation for master equations which describe a stochastic gene regulatory network.