Exact results in the large system size limit for the dynamics of the Chemical Master Equation, a one dimensional chain of equations

TL;DR

This paper uses the Hamilton-Jacobi equation formalism to derive exact analytical solutions for the probability distribution and variance dynamics of the Chemical Master Equation in the large system size limit, including cases with time-dependent rates.

Contribution

It introduces an HJE-based approach to solve the CME analytically in the large system size limit, extending previous methods and providing explicit variance dynamics and criteria for bi-stability disappearance.

Findings

Exact probability distribution expressions derived

Variance dynamics explicitly formulated

Criteria for bi-stability disappearance established

Abstract

We apply the Hamilton-Jacobi equation (HJE) formalism to solve the dynamics of the Chemical Master Equation (CME). We found exact analytical expressions (in large system-size limit) for the probability distribution, including explicit expression for the dynamics of variance of distribution. We also give the solution for some simple cases of the model with time-dependent rates. We derived the results of Van Kampen method from HJE approach using a special ansatz. Using the Van Kampen method, we give a system of ODE to define the variance in 2-d case. We performed numerics for the CME with stationary noise. We give analytical criteria for the disappearance of bi-stability in case of stationary noise in 1-d CME.