Realm of validity of the Crooks relation

Abstract

We consider the distribution P (φ) of the Hatano-Sasa entropy, φ, in reversible and irreversible processes, finding that the Crook's 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 Crook's 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 Crook's relation despite their irreversibility and violation of detailed balance. We verify these results in selected cases numerically.

Entropy production and large deviation function for systems with microscopically irreversible transitions

“…Here we apply the present method to a three-state irreversible loop [20] where the transitions between the three states denoted as 1, 2 and 3 happen in a cyclic way as 1 → 2, 2 → 3, 3 → 1 with rate 1. In the small time interval τ , the transition ratesω ij s have the form,ω 12 =ω 23 =ω 31 ≈ 1 − 2τ, (37) ω 21 =ω 32 =ω 13 ≈ τ.…”

“…The symmetry property of the LDF, I(σ) − I(−σ) = − Ṡ m σ, implies that the fluctuation relation for the entropy production in the medium holds for the system in the long time limit. The plot of the LDF for the entropy production (see figure 6(a)) shows a kink at zero entropy production as a consequence of the fluctuation theorem [17,20]. The distribution function for the entropy production, as shown in 6(b), is obtained directly from equation (47).…”

“…In these systems when some of the transition rates ω ij become zero, this formula predicts an infinite entropy production which have not been observed in realistic situations [18,19]. To address this shortcoming, ben-Avraham et al [20] proposed a regularization scheme by sampling the states of the system at small time interval and obtained the modified transition probabilities. Using those transition probabilities, they computed the medium entropy production rate and its LDF for a three-state irreversible loop.…”

“…These new rates are used to obtain the average rate of medium entropy production and further to obtain the LDF for the entropy production through a saddle point approximation. At zero entropy production, the LDFs for both the models show a kink which can be characterized by the average entropy production rate [17,20,22,23]. Finally, the LDF is used to find the distribution of the entropy production in the long time limit.…”

Entropy production in nonequilibrium steady states: A different approach and an exactly solvable canonical model

“…The time gap between consecutive points on the trajectory is constant τ i = τ . The system can be found in the same state on consecutive samplings, and it could also visit any number of states in between X i and X i+1 [17]. One should note that this interval sampling is readily accessible in experiments, where one usually cannot record every transition between states, as would be needed for event sampling.…”