Conditional reversibility in nonequilibrium stochastic systems

TL;DR

This paper proves that in nonequilibrium Markovian systems, trajectories conditioned on opposite entropy production are related by time reversal, extending the concept of conditional reversibility to stochastic systems.

Contribution

It establishes a stochastic counterpart to Gallavotti's deterministic conditional reversibility theorem, linking entropy-conditioned trajectories through time reversal in Markovian systems.

Findings

Trajectories conditioned on opposite entropy production are time-reversals of each other.

The result applies in the long-time limit for Markovian stochastic systems.

An example illustrates the theoretical findings.

Abstract

For discrete-state stochastic systems obeying Markovian dynamics, we establish the counterpart of the conditional reversibility theorem obtained by Gallavotti for deterministic systems [Ann. de l'Institut Henri Poincar\'e (A) 70, 429 (1999)]. Our result states that stochastic trajectories conditioned on opposite values of entropy production are related by time reversal, in the long-time limit. In other words, the probability of observing a particular sequence of events, given a long trajectory with a specified entropy production rate $\sigma$, is the same as the probability of observing the time-reversed sequence of events, given a trajectory conditioned on the opposite entropy production, $-\sigma$, where both trajectories are sampled from the same underlying Markov process. To obtain our result, we use an equivalence between conditioned ("microcanonical") and biased ("canonical") ensembles of nonequilibrium trajectories. We provide an example to illustrate our findings.