Stochastic thermodynamics of Langevin systems under time-delayed feedback control. II. Nonequilibrium steady-state fluctuations

Abstract

This paper is the second in a series devoted to the study of Langevin systems subjected to a continuous time-delayed feedback control. The goal of our previous paper [Phys. Rev. E 91, 042114 (2015)] was to derive second-law-like inequalities that provide bounds to the average extracted work. Here we study stochastic fluctuations of time-integrated observables such as the heat exchanged with the environment, the extracted work, or the (apparent) entropy production. We use a path-integral formalism and focus on the long-time behavior in the stationary cooling regime, stressing the role of rare events. This is illustrated by a detailed analytical and numerical study of a Langevin harmonic oscillator driven by a linear feedback.

Heat flow due to time-delayed feedback

“…Remarkably, the apparent offset of k B b/γ is independent of the details of the potential landscape. We note that the apparent offset has already been observed and discussed in the context of linear systems [27][28][29][30] . In 30 it has been considered as a consequence of inconsistent usage of Ito and Stratonovich calculus.…”

“…Recent research [27][28][29][30] has revealed that ST of delayed continuous systems is indeed quite involved, even in the absence of nonlinearities. For linear cases, it has been explicitly shown that in the long-time limit a non-equilibrium steady state (NESS) with finite entropy production 30 is approached (in the absence of time-dependent forces), as the delay pushes the system out of equilibrium.…”

“…Nonequilibrium inequalities have been found 28 that generalize the second law and provide bounds to the extractable work. Furthermore, the fluctuations of work, heat and entropy production have been investigated 29 . One crucial issue in the thermodynamic description of delayed systems is the acausality of time-reversed processes appearing in the path integral representation of fluctuating heat q and entropy production [27][28][29] .…”

“…Furthermore, the fluctuations of work, heat and entropy production have been investigated 29 . One crucial issue in the thermodynamic description of delayed systems is the acausality of time-reversed processes appearing in the path integral representation of fluctuating heat q and entropy production [27][28][29] . Contrary to the Markovian case 2 , the total average entropy production ∆S tot of a delayed system in a NESS differs from the medium entropy ∆S m T = q ss (where T is the heat bath's temperature and .. ss denotes NESS ensemble averages).…”

“…Contrary to the Markovian case 2 , the total average entropy production ∆S tot of a delayed system in a NESS differs from the medium entropy ∆S m T = q ss (where T is the heat bath's temperature and .. ss denotes NESS ensemble averages). In particular, the second law does not impose nonnegativity on ∆S m alone [27][28][29] . However, while these statements are generic, explicit expressions for the thermodynamic quantities are, so far, only available for systems governed by linear forces [27][28][29][30] , thus excluding wide classes of physically interesting processes which exclusively arise in nonlinear systems.…”