{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,25]],"date-time":"2026-06-25T13:57:04Z","timestamp":1782395824086,"version":"3.54.5"},"reference-count":84,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2017,12,21]],"date-time":"2017-12-21T00:00:00Z","timestamp":1513814400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100004875","name":"DFG","doi-asserted-by":"publisher","award":["BR 4164\/1-1"],"award-info":[{"award-number":["BR 4164\/1-1"]}],"id":[{"id":"10.13039\/100004875","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100004875","name":"DFG","doi-asserted-by":"publisher","award":["KR 3844\/2-1"],"award-info":[{"award-number":["KR 3844\/2-1"]}],"id":[{"id":"10.13039\/100004875","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100017325","name":"ERC","doi-asserted-by":"publisher","award":["678082"],"award-info":[{"award-number":["678082"]}],"id":[{"id":"10.13039\/100017325","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Living organisms from single cells to humans need to adapt continuously to respond to changes in their environment. The process of behavioural adaptation can be thought of as improving decision-making performance according to some utility function. Here, we consider an abstract model of organisms as decision-makers with limited information-processing resources that trade off between maximization of utility and computational costs measured by a relative entropy, in a similar fashion to thermodynamic systems undergoing isothermal transformations. Such systems minimize the free energy to reach equilibrium states that balance internal energy and entropic cost. When there is a fast change in the environment, these systems evolve in a non-equilibrium fashion because they are unable to follow the path of equilibrium distributions. Here, we apply concepts from non-equilibrium thermodynamics to characterize decision-makers that adapt to changing environments under the assumption that the temporal evolution of the utility function is externally driven and does not depend on the decision-maker\u2019s action. This allows one to quantify performance loss due to imperfect adaptation in a general manner and, additionally, to find relations for decision-making similar to Crooks\u2019 fluctuation theorem and Jarzynski\u2019s equality. We provide simulations of several exemplary decision and inference problems in the discrete and continuous domains to illustrate the new relations.<\/jats:p>","DOI":"10.3390\/e20010001","type":"journal-article","created":{"date-parts":[[2017,12,21]],"date-time":"2017-12-21T12:16:14Z","timestamp":1513858574000},"page":"1","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Non-Equilibrium Relations for Bounded Rational Decision-Making in Changing Environments"],"prefix":"10.3390","volume":"20","author":[{"given":"Jordi","family":"Grau-Moya","sequence":"first","affiliation":[{"name":"Max Planck Institute for Intelligent Systems, Stuttgart 70569, Germany"},{"name":"Max Planck Institute for Biological Cybernetics, T\u00fcbingen 72076, Germany"},{"name":"PROWLER.io, Cambridge CB2 1LA, UK"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Matthias","family":"Kr\u00fcger","sequence":"additional","affiliation":[{"name":"Max Planck Institute for Intelligent Systems, Stuttgart 70569, Germany"},{"name":"4th Institute for Theoretical Physics, Universit\u00e4t Stuttgart, Stuttgart 70569, Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8637-6652","authenticated-orcid":false,"given":"Daniel","family":"Braun","sequence":"additional","affiliation":[{"name":"Max Planck Institute for Intelligent Systems, Stuttgart 70569, Germany"},{"name":"Max Planck Institute for Biological Cybernetics, T\u00fcbingen 72076, Germany"},{"name":"Institute of Neural Information Processing, Universit\u00e4t Ulm, Ulm 89081, Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2017,12,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Ortega, P.A., and Braun, D.A. 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