Solution formula for the general birth-death chemical diffusion master equation

TL;DR

This paper introduces a solution formula for birth-death chemical diffusion master equations, linking stochastic molecular diffusion models with reaction-diffusion PDEs, and demonstrating its effectiveness through examples.

Contribution

It provides a novel solution formula connecting chemical diffusion master equations with reaction-diffusion PDEs, enhancing analytical tools for spatially distributed chemical systems.

Findings

Solution formula expressed via reaction-diffusion PDEs

Analogy with classical birth-death master equations

Illustrated with multiple example systems

Abstract

We propose a solution formula for chemical diffusion master equations of birth and death type. These equations, proposed and formalized in the recent paper [5], aim at incorporating the spatial diffusion of molecules into the description provided by the classical chemical master equation. We start from the general approach developed in [20] and perform a more detailed analysis of the representation found there. This leads to a solution formula for birth-death chemical diffusion master equations which is expressed in terms of the solution to the reaction-diffusion partial differential equation associated with the system under investigation. Such representation also reveals a striking analogy with the solution to the classical birth-death chemical master equations. The solutions of our findings are also illustrated for several examples.