Universal interpretation of efficacy parameter in perturbed nonequilibrium systems

The fluctuation theorems have remained one of the cornerstones in the study of systems that are driven far out of equilibrium, and they provide strong constraints on the fraction of trajectories that behave atypically in light of the second law. They have mainly been derived for a predetermined external drive applied to the system. However, to improve the efficiency of a process, one needs to incorporate protocols that are modified by receiving feedbacks about the recent state of the system, during its evolution. In such a case, the forms of the conventional fluctuation theorems get modified, the correction term involving terms that depend on the way the reverse/conjugate process is defined, namely, the rules of using feedback in order to generate the exact time-reversed/conjugate protocols. We show in this paper that this can be done in a large number of ways, and in each case we would get a different expression for the correction terms. This would in turn lead to several lower bounds on the mean work that must be performed on the system, or on the entropy changes. Here we analyze a form of the extended fluctuation theorems that involves the efficacy parameter, and find that this form retains a consistent physical meaning regardless of the design of feedback along the conjugate process, as opposed to the case of the previously mentioned form of the modified fluctuation theorems.