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Entropy Production in Exactly Solvable Systems
Article

Medium Entropy Reduction and Instability in Stochastic Systems with Distributed Delay

1
Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstr. 36, 10623 Berlin, Germany
2
ICTP—The Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, 34151 Trieste, Italy
3
Institut für Theoretische Physik, Universität Leipzig, Brüderstraße 15, 04103 Leipzig, Germany
4
Institut für Physik, Humboldt-Universität zu Berlin, Newtonstr. 15, 12489 Berlin, Germany
*
Author to whom correspondence should be addressed.
Academic Editor: Henni Ouerdane
Entropy 2021, 23(6), 696; https://doi.org/10.3390/e23060696
Received: 3 May 2021 / Revised: 20 May 2021 / Accepted: 26 May 2021 / Published: 31 May 2021
(This article belongs to the Special Issue Nonequilibrium Thermodynamics and Stochastic Processes)
Many natural and artificial systems are subject to some sort of delay, which can be in the form of a single discrete delay or distributed over a range of times. Here, we discuss the impact of this distribution on (thermo-)dynamical properties of time-delayed stochastic systems. To this end, we study a simple classical model with white and colored noise, and focus on the class of Gamma-distributed delays which includes a variety of distinct delay distributions typical for feedback experiments and biological systems. A physical application is a colloid subject to time-delayed feedback control, which is, in principle, experimentally realizable by co-moving optical traps. We uncover several unexpected phenomena in regard to the system’s linear stability and its thermodynamic properties. First, increasing the mean delay time can destabilize or stabilize the process, depending on the distribution of the delay. Second, for all considered distributions, the heat dissipated by the controlled system (e.g., the colloidal particle) can become negative, which implies that the delay force extracts energy and entropy of the bath. As we show here, this refrigerating effect is particularly pronounced for exponential delay. For a specific non-reciprocal realization of a control device, we find that the entropic costs, measured by the total entropy production of the system plus controller, are the lowest for exponential delay. The exponential delay further yields the largest stable parameter regions. In this sense, exponential delay represents the most effective and robust type of delayed feedback. View Full-Text
Keywords: non-Markovian dynamics; stochastic thermodynamics; time-delayed feedback control; stochastic delay differential equations; feedback cooling; non-reciprocal interactions non-Markovian dynamics; stochastic thermodynamics; time-delayed feedback control; stochastic delay differential equations; feedback cooling; non-reciprocal interactions
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MDPI and ACS Style

Loos, S.A.M.; Hermann, S.; Klapp, S.H.L. Medium Entropy Reduction and Instability in Stochastic Systems with Distributed Delay. Entropy 2021, 23, 696. https://doi.org/10.3390/e23060696

AMA Style

Loos SAM, Hermann S, Klapp SHL. Medium Entropy Reduction and Instability in Stochastic Systems with Distributed Delay. Entropy. 2021; 23(6):696. https://doi.org/10.3390/e23060696

Chicago/Turabian Style

Loos, Sarah A.M., Simon Hermann, and Sabine H.L. Klapp 2021. "Medium Entropy Reduction and Instability in Stochastic Systems with Distributed Delay" Entropy 23, no. 6: 696. https://doi.org/10.3390/e23060696

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