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

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