Lyapunov functions, stationary distributions, and non-equilibrium potential for chemical reaction networks

TL;DR

This paper explores the connection between stationary distributions of stochastic reaction models and Lyapunov functions of their deterministic versions, revealing new ways to derive Lyapunov functions from stationary distributions.

Contribution

It derives Lyapunov functions as scaling limits of non-equilibrium potentials, extending the approach to general birth-death models with multiple equilibria.

Findings

Lyapunov functions can be obtained from stationary distributions via scaling limits

The approach applies to complex balanced and more general birth-death models

Lyapunov functions can be constructed even for models with multiple equilibria

Abstract

We consider the relationship between stationary distributions for stochastic models of reaction systems and Lyapunov functions for their deterministic counterparts. Specifically, we derive the well known Lyapunov function of reaction network theory as a scaling limit of the non-equilibrium potential of the stationary distribution of stochastically modeled complex balanced systems. We extend this result to general birth-death models and demonstrate via example that similar scaling limits can yield Lyapunov functions even for models that are not complex or detailed balanced, and may even have multiple equilibria.