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Lyapunov Functions, Stationary Distributions, and Non-equilibrium Potential for Reaction Networks
David F Anderson1, Gheorghe Craciun2,3, Manoj Gopalkrishnan4
1Department of Mathematics, University of Wisconsin-Madison, Madison, WI, USA. anderson@math.wisc.edu.
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.
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