Fluctuation Theorems for Entropy Production and Heat Dissipation in Periodically Driven Markov Chains

TL;DR

This paper extends fluctuation theorems to periodically driven Markov chains without stationary distributions, revealing a generalized symmetry relating forward and backward processes and providing new insights into entropy production and heat dissipation.

Contribution

It introduces a novel fluctuation theorem for inhomogeneous Markov processes driven periodically, generalizing existing results to non-stationary systems.

Findings

Generalized Gallavotti-Cohen symmetry for periodic Markov chains

Relation between fluctuation distributions in forward and backward processes

Expression of entropy production as an integral of a periodic rate

Abstract

Asymptotic fluctuation theorems are statements of a Gallavotti-Cohen symmetry in the rate function of either the time-averaged entropy production or heat dissipation of a process. Such theorems have been proved for various general classes of continuous-time deterministic and stochastic processes, but always under the assumption that the forces driving the system are time independent, and often relying on the existence of a limiting ergodic distribution. In this paper we extend the asymptotic fluctuation theorem for the first time to inhomogeneous continuous-time processes without a stationary distribution, considering specifically a finite state Markov chain driven by periodic transition rates. We find that for both entropy production and heat dissipation, the usual Gallavotti-Cohen symmetry of the rate function is generalized to an analogous relation between the rate functions of the original process and its corresponding backward process, in which the trajectory and the driving protocol have been time-reversed. The effect is that spontaneous positive fluctuations in the long time average of each quantity in the forward process are exponentially more likely than spontaneous negative fluctuations in the backward process, and vice-versa, revealing that the distributions of fluctuations in universes in which time moves forward and backward are related. As an additional result, the asymptotic time-averaged entropy production is obtained as the integral of a periodic entropy production rate that generalizes the constant rate pertaining to homogeneous dynamics.