Monotone return to steady nonequilibrium

TL;DR

This paper introduces a new Lyapunov function based on large deviation principles that monotonically approaches nonequilibrium steady states, contrasting with entropy-based functions that oscillate far from equilibrium.

Contribution

It proposes a novel Lyapunov functional derived from fluctuation asymptotics, providing a more reliable measure of relaxation to nonequilibrium steady states.

Findings

The functional measures excess dynamical activity rates in driven processes.

Numerical evidence supports its monotonic convergence near steady states.

Rigorous arguments confirm its decreasing behavior over time.

Abstract

We propose and analyze a new candidate Lyapunov function for relaxation towards general nonequilibrium steady states. The proposed functional is obtained from the large time asymptotics of time-symmetric fluctuations. For driven Markov jump or diffusion processes it measures an excess in dynamical activity rates. We present numerical evidence and we report on a rigorous argument for its monotonous time-dependence close to the steady nonequilibrium or in general after a long enough time. This is in contrast with the behavior of approximate Lyapunov functions based on entropy production that when driven far from equilibrium often keep exhibiting temporal oscillations even close to stationarity.