Monotonicity of the dynamical activity

TL;DR

This paper proves that the Donsker-Varadhan rate function, related to dynamical activity in Markov processes, monotonically approaches stationarity over time under certain conditions, extending understanding of nonequilibrium fluctuations.

Contribution

It provides a rigorous proof of monotonicity of the Donsker-Varadhan rate function under Markov evolution for large times, given a normal linear-response condition.

Findings

Donsker-Varadhan function is monotone at large times

Monotonicity holds under a normal linear-response condition

Supports numerical observations of monotone return to stationarity

Abstract

The Donsker-Varadhan rate function for occupation-time fluctuations has been seen numerically to exhibit monotone return to stationary nonequilibrium [Phys. Rev. Lett. 107, 010601 (2011)]. That rate function is related to dynamical activity and, except under detailed balance, it does not derive from the relative entropy for which the monotonicity in time is well understood. We give a rigorous argument that the Donsker-Varadhan function is indeed monotone under the Markov evolution at large enough times with respect to the relaxation time, provided that a "normal linear-response" condition is satisfied.