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

David F. Anderson, Gheorghe Craciun, Manoj Gopalkrishnan, Carsten Wiuf

Published 2014-10-17Version 1

We consider the relationship between stationary distributions for stochastic models of chemical reaction systems and Lyapunov functions for their deterministic counterparts. Specifically, we derive the well known Lyapunov function of chemical 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.