{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,5]],"date-time":"2026-02-05T07:23:02Z","timestamp":1770276182505,"version":"3.49.0"},"reference-count":90,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2021,5,31]],"date-time":"2021-05-31T00:00:00Z","timestamp":1622419200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["Projektnummer 163436311 - SFB 910"],"award-info":[{"award-number":["Projektnummer 163436311 - SFB 910"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>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\u2019s 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.<\/jats:p>","DOI":"10.3390\/e23060696","type":"journal-article","created":{"date-parts":[[2021,5,31]],"date-time":"2021-05-31T23:33:33Z","timestamp":1622504013000},"page":"696","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Medium Entropy Reduction and Instability in Stochastic Systems with Distributed Delay"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5946-5684","authenticated-orcid":false,"given":"Sarah A. M.","family":"Loos","sequence":"first","affiliation":[{"name":"Institut f\u00fcr Theoretische Physik, Technische Universit\u00e4t Berlin, Hardenbergstr. 36, 10623 Berlin, Germany"},{"name":"ICTP\u2014The Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, 34151 Trieste, Italy"},{"name":"Institut f\u00fcr Theoretische Physik, Universit\u00e4t Leipzig, Br\u00fcderstra\u00dfe 15, 04103 Leipzig, Germany"}]},{"given":"Simon","family":"Hermann","sequence":"additional","affiliation":[{"name":"Institut f\u00fcr Physik, Humboldt-Universit\u00e4t zu Berlin, Newtonstr. 15, 12489 Berlin, Germany"}]},{"given":"Sabine H. L.","family":"Klapp","sequence":"additional","affiliation":[{"name":"Institut f\u00fcr Theoretische Physik, Technische Universit\u00e4t Berlin, Hardenbergstr. 36, 10623 Berlin, Germany"}]}],"member":"1968","published-online":{"date-parts":[[2021,5,31]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Sch\u00f6ll, E., Klapp, S.H.L., and H\u00f6vel, P. (2016). Control of Self-Organizing Nonlinear Systems, Springer.","DOI":"10.1007\/978-3-319-28028-8"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"783","DOI":"10.1103\/RevModPhys.77.783","article-title":"Feedback for physicists: A tutorial essay on control","volume":"77","author":"Bechhoefer","year":"2005","journal-title":"Rev. Mod. Phys."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Sch\u00f6ll, E., and Schuster, H.G. (2008). 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