Abstract
The thermodynamic uncertainty relation originally proven for systems driven into a non-equilibrium steady state (NESS) allows one to infer the total entropy production rate by observing any current in the system. This kind of inference scheme is especially useful when the system contains hidden degrees of freedom or hidden discrete states, which are not accessible to the experimentalist. A recent generalization of the thermodynamic uncertainty relation to arbitrary time-dependent driving allows one to infer entropy production not only by measuring current-observables but also by observing state variables. A crucial question then is to understand which observable yields the best estimate for the total entropy production. In this paper we address this question by analyzing the quality of the thermodynamic uncertainty relation for various types of observables for the generic limiting cases of fast driving and slow driving. We show that in both cases observables can be found that yield an estimate of order one for the total entropy production. We further show that the uncertainty relation can even be saturated in the limit of fast driving.
Highlights
Recent progresses in the field of non-equilibrium statistical physics have reshaped our perspective on conventional thermodynamic notions such as work, heat or entropy production
Milestones in the field of stochastic thermodynamics inter alia deal with similar connections for systems far away from equilibrium: from fluctuation theorems [6,7,8,9,10,11,12,13,14,15] and generalizations of the fluctuation-dissipation theorem (FDT) [16,17,18,19,20] to the Harada-Sasa relation connecting the violation of the FDT to energy dissipation [21, 22]
We have analyzed the quality of the thermodynamic uncertainty relation for the limiting cases of fast driving and slow driving
Summary
Recent progresses in the field of non-equilibrium statistical physics have reshaped our perspective on conventional thermodynamic notions such as work, heat or entropy production. A unifying uncertainty relation for arbitrary timedependent driving including the TUR for finite observation times [25,26], for relaxation processes [64,65] and for periodically driven systems [47] has been found recently [66] A similar inequality involving the total entropy production rate can be derived for state variables [66] Since their origin lies in the response of the system with respect to a time re-scaling by using a virtual perturbing force [67], these relations should be clearly distinguished from so-called generalized thermodynamic uncertainty relations that are solely a consequence of the fluctuation theorem [68, 69]. We show that these results hold for systems with continuous degrees of freedom, and for systems with a discrete set of states as we illustrate for a driven three-state model
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