Linear response in large deviations theory: a method to compute non-equilibrium distributions

Abstract

We consider thermodynamically consistent autonomous Markov jump processes displaying a macroscopic limit in which the logarithm of the probability distribution is proportional to a scale-independent rate function (i.e. a large deviations principle is satisfied). In order to provide an explicit expression for the probability distribution valid away from equilibrium, we propose a linear response theory performed at the level of the rate function. We show that the first order non-equilibrium contribution to the steady state rate function, g ( x ), satisfies $\boldsymbol{u}(\boldsymbol{x})\cdot \nabla g(\boldsymbol{x})=-\beta \dot {W}(\boldsymbol{x})$ where the vector field u ( x ) defines the macroscopic deterministic dynamics, and the scalar field $\dot {W}(\boldsymbol{x})$ equals the rate at which work is performed on the system in a given state x . This equation provides a practical way to determine g ( x ), significantly outperforms standard linear response theory applied at the level of the probability distribution, and approximates the rate function surprisingly well in some far-from-equilibrium conditions. The method applies to a wealth of physical and chemical systems, that we exemplify by two analytically tractable models—an electrical circuit and an autocatalytic chemical reaction network—both undergoing a non-equilibrium transition from a monostable phase to a bistable phase. Our approach can be easily generalized to transient probabilities and non-autonomous dynamics. Moreover, its recursive application generates a virtual flow in the probability space which allows to determine the steady state rate function arbitrarily far from equilibrium.

Keywords

• large deviations theory
• linear response
• stochastic thermodynamics
• non-equilibrium steady states