Lyapunov Functions, Stationary Distributions, and Non-equilibrium Potential for Reaction Networks

Area of Science:

• Chemical kinetics
• Stochastic modeling
• Dynamical systems theory

Background:

• Reaction network theory often uses deterministic models with Lyapunov functions to analyze stability.
• Stochastic models offer a more detailed view of reaction dynamics, especially at low molecule numbers.
• Connecting these two modeling approaches is crucial for a comprehensive understanding of reaction systems.

Purpose of the Study:

• To investigate the relationship between stationary distributions in stochastic reaction models and Lyapunov functions in their deterministic counterparts.
• To derive the Lyapunov function of reaction network theory from the stationary distribution of stochastic models.
• To extend this derivation to broader classes of reaction models.

Main Methods:

• Analyzing stationary distributions of stochastic reaction systems.
• Utilizing non-equilibrium potential concepts.
• Applying scaling limit techniques.
• Examining birth-death models and other reaction systems.

Main Results:

• The Lyapunov function from reaction network theory is derived as a scaling limit of the non-equilibrium potential of stationary distributions for complex balanced systems.
• This derivation is successfully extended to general birth-death models.
• The method demonstrates the ability to yield Lyapunov functions for systems that are not complex or detailed balanced and may possess multiple equilibria.

Conclusions:

• A fundamental link is established between stochastic stationary distributions and deterministic Lyapunov functions in reaction systems.
• The derived scaling limit method provides a powerful tool for analyzing the stability of a wider range of reaction models.
• This work bridges stochastic and deterministic perspectives in chemical kinetics and dynamical systems theory.