Realm of Validity of the Crooks Relation

TL;DR

This paper demonstrates that the Crooks relation applies not only to reversible processes but also to certain irreversible, slow processes, especially in systems with finite states, including some that violate detailed balance.

Contribution

It extends the validity of the Crooks relation to irreversible processes in the adiabatic limit and identifies classes of multi-state systems that obey it through coarse-graining.

Findings

Crooks relation holds for slow irreversible processes.

Two-state systems always satisfy Crooks relation and detailed balance.

Certain multi-state systems obey Crooks relation via coarse-graining.

Abstract

We consider the distribution $P(\phi)$ of the Hatano-Sasa entropy, $\phi$, in reversible and irreversible processes, finding that the Crooks relation for the ratio of the pdf's of the forward and backward processes, $P_F(\phi)/P_R(-\phi)=e^{\phi}$, is satisfied not only for reversible, but also for irreversible processes, in general, in the adiabatic limit of "slow processes." Focusing on systems with a finite set of discrete states (and no absorbing states), we observe that two-state systems always fulfill detailed balance, and obey Crooks relation. We also identify a wide class of systems, with more than two states, that can be "coarse-grained" into two-state systems and obey Crooks relation despite their irreversibility and violation of detailed balance. We verify these results in selected cases numerically.