Non-equilibrium Lyapunov function and a fluctuation relation for stochastic systems: Poisson representation approach

TL;DR

This paper develops a statistical physics framework using the Poisson representation to analyze nonlinear non-equilibrium stochastic processes, deriving a fluctuation relation and identifying a Lyapunov function that monotonically decreases over time.

Contribution

It introduces a novel approach combining the Poisson representation with large-deviation principles to derive fluctuation relations and non-equilibrium Lyapunov functions for complex stochastic systems.

Findings

Derivation of an integral fluctuation relation for nonlinear systems under feedback.

Identification of a non-equilibrium Lyapunov function that decays monotonically.

Application of the method to biophysical processes like bacterial chemosensing.

Abstract

We present a statistical physics framework for description of nonlinear non-equilibrium stochastic processes, modeled via chemical master equation, in the weak-noise limit. Using the Poisson representation approach and applying the large-deviation principle we first solve the master equation. Then we use the notion of the non-equilibrium free energy to derive an integral fluctuation relation for nonlinear non-equilibrium systems under feedback control. We point out that the free energy as well as some functionals can serve as non-equilibrium Lyapunov function which has an important property to decay to its minimal value monotonously at all times. The Poisson representation technique is illustrated via exact stochastic treatment of biophysical processes, such as bacterial chemosensing and molecular evolution.