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Realm of Validity of the Crooks Relation

We consider the distribution $P(ϕ)$ of the Hatano-Sasa entropy, $ϕ$, in reversible and irreversible processes, finding that the Crooks relation for the ratio of the pdf's of the forward and backward processes, $P_F(ϕ)/P_R(-ϕ)=e^ϕ$, 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.