{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,15]],"date-time":"2026-06-15T14:48:39Z","timestamp":1781534919421,"version":"3.54.5"},"reference-count":57,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2026,4,11]],"date-time":"2026-04-11T00:00:00Z","timestamp":1775865600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12564032"],"award-info":[{"award-number":["12564032"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"award":["12564032"],"award-info":[{"award-number":["12564032"]}],"id":[{"id":"https:\/\/ror.org\/01h0zpd94","id-type":"ROR","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100019413","name":"Yunnan Provincial Department of Education Science Research Fund Project","doi-asserted-by":"crossref","award":["2025J0942"],"award-info":[{"award-number":["2025J0942"]}],"id":[{"id":"10.13039\/501100019413","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100019413","name":"Yunnan Provincial Department of Education Science Research Fund Project","doi-asserted-by":"crossref","award":["2026J0977"],"award-info":[{"award-number":["2026J0977"]}],"id":[{"id":"10.13039\/501100019413","id-type":"DOI","asserted-by":"crossref"}]},{"name":"Yunnan Provincial Xiao Rui Expert Workstation","award":["202605AF350031"],"award-info":[{"award-number":["202605AF350031"]}]},{"name":"2025 Self-funded Science and Technology Projects of Chuxiong Prefecture","award":["cxzc2025004"],"award-info":[{"award-number":["cxzc2025004"]}]},{"name":"2025 Self-funded Science and Technology Projects of Chuxiong Prefecture","award":["cxzc2025008"],"award-info":[{"award-number":["cxzc2025008"]}]},{"name":"2025 Self-funded Science and Technology Projects of Chuxiong Prefecture","award":["cxzc2025014"],"award-info":[{"award-number":["cxzc2025014"]}]},{"name":"Chuxiong Normal University Doctoral Research Initiation Fund Project","award":["BSQD2407"],"award-info":[{"award-number":["BSQD2407"]}]},{"name":"Chuxiong Normal University Doctoral Research Initiation Fund Project","award":["BSQD2507"],"award-info":[{"award-number":["BSQD2507"]}]},{"name":"Dongying Science Development Fund","award":["DJB2023015"],"award-info":[{"award-number":["DJB2023015"]}]},{"name":"Independent Research Fund of Yunnan Tin & Indium Laboratory","award":["202405AR340002-25BC01"],"award-info":[{"award-number":["202405AR340002-25BC01"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Living systems are open nonequilibrium systems that continuously exchange energy, matter, and information with their environments, leading to stochastic dynamics with memory and active fluctuations. In this study, we derive a microscopic non-Markovian description of entropy dynamics in living systems using the Keldysh functional formalism, providing a quantitative foundation for the entropy bathtub picture. The approach naturally incorporates colored environmental noise, memory-dependent dissipation, and many-body interactions, yielding generalized Langevin dynamics and non-Markovian master equations. Within this framework we derive an exact frequency-domain expression for the entropy production rate and show that violations of the fluctuation\u2013dissipation relation provide a direct thermodynamic signature of active biological fluctuations. We further demonstrate that environmental memory enhances low-frequency fluctuations and entropy production, leading to critical slowing down near dynamical instability. These results provide a microscopic physical foundation for the entropy \u201cbathtub\u201d picture of living systems and connect entropy evolution with development, aging, and death in nonequilibrium dynamics.<\/jats:p>","DOI":"10.3390\/e28040428","type":"journal-article","created":{"date-parts":[[2026,4,13]],"date-time":"2026-04-13T10:19:14Z","timestamp":1776075554000},"page":"428","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Non-Markovian Entropy Dynamics in Living Systems from the Keldysh Formalism"],"prefix":"10.3390","volume":"28","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2074-669X","authenticated-orcid":false,"given":"Feiyi","family":"Liu","sequence":"first","affiliation":[{"name":"School of Physics, Electrical and Energy Engineering, Chuxiong Normal University, Chuxiong 675000, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0004-9016-9615","authenticated-orcid":false,"given":"Min","family":"Guo","sequence":"additional","affiliation":[{"name":"School of Physics, Electrical and Energy Engineering, Chuxiong Normal University, Chuxiong 675000, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9862-1643","authenticated-orcid":false,"given":"Hongwei","family":"Tan","sequence":"additional","affiliation":[{"name":"School of Science, Hunan Institute of Technology, Hengyang 421002, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3217-5775","authenticated-orcid":false,"given":"Yang","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Information Science and Engineering, Shandong Institute of Petroleum and Chemical Technology, Dongying 257061, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2026,4,11]]},"reference":[{"key":"ref_1","unstructured":"Nicolis, G., and Prigogine, I. 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