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

TL;DR

This paper introduces a linear response method at the level of large deviations rate functions to efficiently compute non-equilibrium distributions in Markov jump processes, outperforming traditional approaches especially far from equilibrium.

Contribution

The authors develop a linear response theory for rate functions that provides explicit formulas for non-equilibrium steady states, extending the applicability beyond near-equilibrium conditions.

Findings

The method accurately approximates rate functions in far-from-equilibrium regimes.

It outperforms standard linear response theory applied to probability distributions.

The approach is validated on models like electrical circuits and chemical reactions.

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 $u(x)\cdot \nabla g(x) = -\beta \dot W(x)$ where the vector field $u(x)$ defines the macroscopic deterministic dynamics, and the scalar field $\dot W(x)$ 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.