Time-reversal symmetric Crooks and Gallavotti–Cohen fluctuation relations in driven classical Markovian systems

Abstract

In this paper, we address an important question of the relationship between fluctuation theorems for the dissipated work W d = W −∆F with general finitetime (like Jarzynski equality and Crooks relation) and infinite-time (like Gallavotti-Cohen theorem) drive protocols and their time-reversal symmetric versions. The relations between these kinds of fluctuation relations are uncovered based on the examples of a classical Markovian N -level system. Further consequences of these relations are discussed with respect to the possible experimental verifications.

Fluctuations in heat engines

“…However, it is important to note that an arbitrary protocol can be approximated by a piece-wise constant driving [43,137] and thus the following approach can be applied for extracting information on work and heat fluctuations in an arbitrarily driven two-level system described by the GME (1). Moreover, the key steps of the presented technique do not change with increasing number of microstates and thus it applies for arbitrary discrete systems [135]. Finally, reference [43] shows that the overdamped continuous systems (8) can be well approximated by discrete systems and thus the presented method yields work and heat fluctuations also for overdamped Brownian heat engines.…”

“…Exact solutions for specific piece-wise constant protocols were obtained in references [75,133] and applied in study of stochastic efficiency [77,88], current fluctuations [134], time-reversal symmetric Crooks and Gallavotti-Cohen fluctuation relations [135], and TURs [120]. Moment-generating function for work was also discussed recently in [136].…”

Fluctuations in heat engines

“…Exact solutions for specific piece-wise constant protocols were obtained in Refs. [59,118] and applied in study of fluctuating efficiency [61,72], current fluctuations [119], time-reversal symmetric Crooks and Gallavotti-Cohen fluctuation relations [120], and thermodynamic uncertainty relation [105]. Moment-generating function for work was also discussed recently in [121].…”

“…However, it is important to note that an arbitrary protocol can be approximated by a piecewise constant driving [42,122] and thus the following approach can be applied for extracting information on work and heat fluctuations in arbitrarily driven two-level system described by the GME (1). Moreover, the nature of key steps of the presented technique don't change with increasing number of microstates and thus it applies for arbitrary discrete systems [120]. Finally, Ref.…”

Stochastic thermodynamics: experiment and theory

“…These relations clarify surprising connections between fluctuation relations and topics such as first-passage-time distributions and the fluctuations of waveforms near the Anderson localization transition. [14] • Bo, Lim, and Eichhorn add rigor to the justification of the commonly used Stratonovich convention in evaluating discretized, path-dependent stochastic quantities such as heat and work. Starting from generalized Langevin dynamics driven by noise with a finite time correlation, they carefully consider the whitenoise and small-mass limits to arrive at the commonly used overdamped expressions.…”