Nonequilibrium fluctuation theorem for systems under discrete and continuous feedback control

Abstract

Without violating causality, we allow performing measurements in time reverse process of a feedback manipulated stochastic system. As a result we come across an entropy production due to the measurement process. This entropy production, in addition to the usual system and medium entropy production, constitutes the total entropy production of the combined system of the reservoir, the system and the feedback controller. We show that this total entropy production of "full" system satisfies an integrated fluctuation theorem as well as a detailed fluctuation theorem as expected. We illustrate and verify this idea through explicit calculation and direct simulation in two examples.

Stochastic thermodynamics based on incomplete information: generalized Jarzynski equality with measurement errors with or without feedback

“…. However, in case of feedback at a given time t m Sagawa and Ueda and others have found that [25][26][27][28][29][30][31][32][33][34] ( ) to the exponent we can make the right-hand side of the 'Jarzynski-Sagawa-Ueda relation' equal to unity again. This result provides us with a nice interpretation because it tells us that the amount of work we can extract from the system is bounded by ( ) á ñ I y z , m m z y , m m , which can be viewed as the amount of correlations established during the measurement.…”

“…Feedback describes the situation in which the state of the system is measured and the evolution of the system is manipulated by applying an external control scheme depending on the measurement outcome. The change of the JE and other fluctuation theorems under feedback has recently attracted a lot of attention, in theory [25][26][27][28][29][30][31][32][33][34] as well as in experiments [35,36]. A prominent and the first example of a generalized JE incorporating feedback by performing a single measurement on a stochastic thermodynamic system at a time t m with measurement outcome y m is the relation derived by Sagawa and Ueda [25]:…”

“…but at this point unspecified driving protocol an approximation for the modification of the original JE (see appendix A.6 for details): is the probability for the system to be in state z (ground or excited) at time t in the backward process with the backward protocol according to the measurement outcome y m (ground or excited state) in the forward process. We again note that we do not apply feedback in the backward process and that equation (45) is valid under exactly the same conditions as discussed below equation (28). Furthermore, ( ) † w l t y , m is the energy gap as defined in section 3.3 with the time-reversed protocol according to the outcome of the forward process.…”

Identifying functional thermodynamics in autonomous Maxwellian ratchets

“…Importantly for the following, this Second Law assumes explicitly observed Markov system dynamics [17] and quantifies this relevant information only in terms of the distribution of instantaneous system microstates; not, to emphasize, microstate path entropies. In short, while the system's instantaneous distributions relax and change over time, the information reservoir itself is not allowed to build up and store memory or correlations.Note that this framework of information reservoirs differs from alternative approaches to the thermodynamics of information processing, including: (i) active feedback control by external means, where the thermodynamic account of the Demon's activities tracks the mutual information between measurement outcomes and system state [20][21][22][23][24][25][26][27][28][29][30][31][32][33]; (ii) the multipartite framework where, for a set of interacting, stochastic subsystems, the Second Law is expressed via their intrinsic entropy production, correlations among them, and transfer entropy [34][35][36][37]; and (iii) steady-state models that invoke time-scale separation to identify a portion of the overall entropy production as an information current [38,39]. A unified approach to these perspectives was attempted in [40][41][42].Recently, Maxwellian Demons have been proposed to explore plausible automated mechanisms that appeal to equation (2)ʼs modified Second Law to do useful work, by decreasing the physical entropy, at the expense of positive change in reservoir Shannon information [39,[43][44][45][46][47][48].…”

“…Note that this framework of information reservoirs differs from alternative approaches to the thermodynamics of information processing, including: (i) active feedback control by external means, where the thermodynamic account of the Demon's activities tracks the mutual information between measurement outcomes and system state [20][21][22][23][24][25][26][27][28][29][30][31][32][33]; (ii) the multipartite framework where, for a set of interacting, stochastic subsystems, the Second Law is expressed via their intrinsic entropy production, correlations among them, and transfer entropy [34][35][36][37]; and (iii) steady-state models that invoke time-scale separation to identify a portion of the overall entropy production as an information current [38,39]. A unified approach to these perspectives was attempted in [40][41][42].…”