From chemical Langevin equations to Fokker-Planck equation: application of Hodge decomposition and Klein-Kramers equation

TL;DR

This paper investigates the conditions for potential landscape existence in complex chemical reaction systems modeled by Langevin equations, utilizing Hodge decomposition and deriving the Fokker-Planck equation through Hamiltonian mapping.

Contribution

It introduces a novel application of Hodge decomposition to analyze potential landscapes in chemical Langevin systems and derives the associated Fokker-Planck equation.

Findings

Conditions for potential landscape existence are identified.

Fokker-Planck equation for complex chemical systems is derived.

Framework enables analysis of steady-state properties.

Abstract

The stochastic systems without detailed balance are common in various chemical reaction systems, such as metabolic network systems. In studies of these systems, the concept of potential landscape is useful. However, what are the sufficient and necessary conditions of the existence of the potential function is still an open problem. Use Hodge decomposition theorem in differential form theory, we focus on the general chemical Langevin equations, which reflect complex chemical reaction systems. We analysis the conditions for the existence of potential landscape of the systems. By mapping the stochastic differential equations to a Hamiltonian mechanical system, we obtain the Fokker-Planck equation of the chemical reaction systems. The obtained Fokker-Planck equation can be used in further studies of other steady properties of complex chemical reaction systems, such as their steady state entropies.