{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,30]],"date-time":"2026-07-30T12:14:29Z","timestamp":1785413669145,"version":"3.56.0"},"reference-count":25,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2026,6,9]],"date-time":"2026-06-09T00:00:00Z","timestamp":1780963200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"ANID, Chile","award":["1220651"],"award-info":[{"award-number":["1220651"]}]},{"name":"ANID, Chile","award":["1240281"],"award-info":[{"award-number":["1240281"]}]},{"DOI":"10.13039\/501100020884","name":"Agencia Nacional de Investigaci\u00f3n y Desarrollo","doi-asserted-by":"crossref","award":["21210407"],"award-info":[{"award-number":["21210407"]}],"id":[{"id":"10.13039\/501100020884","id-type":"DOI","asserted-by":"crossref"}]},{"award":["21210407"],"award-info":[{"award-number":["21210407"]}],"id":[{"id":"https:\/\/ror.org\/02ap3w078","id-type":"ROR","asserted-by":"publisher"}]},{"name":"Italian Space Agency","award":["2024-5-E.0\u2013CUP"],"award-info":[{"award-number":["2024-5-E.0\u2013CUP"]}]},{"name":"Italian Space Agency","award":["I53D24000060005"],"award-info":[{"award-number":["I53D24000060005"]}]},{"award":["2024-5-E.0\u2013CUP"],"award-info":[{"award-number":["2024-5-E.0\u2013CUP"]}],"id":[{"id":"https:\/\/ror.org\/034zgem50","id-type":"ROR","asserted-by":"publisher"}]},{"award":["I53D24000060005"],"award-info":[{"award-number":["I53D24000060005"]}],"id":[{"id":"https:\/\/ror.org\/034zgem50","id-type":"ROR","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this work, we formulate a systematic expectation-value framework for dynamical systems whose probability densities evolve according to linear partial differential equations, such as the Fokker-Planck and Liouville equations. The approach is based on expectation-calculus identities associated with the Fluctuation-Dissipation Theorem and the Conjugate Variables Theorem, allowing the derivation of evolution equations directly for arbitrary observables and fluctuations without explicitly solving the full probability-density equation. The resulting relations provide a classical Ehrenfest-type formulation for observable dynamics and fluctuations under linear probability-density evolution. While the resulting equations are not closed in general, since they typically involve higher-order moments, correlations, or derivatives, the formalism offers a unified operational framework for studying observable dynamics under suitable approximations or closure assumptions. We illustrate the procedure with examples involving Fokker\u2013Planck and Liouville dynamics and discuss the scope, limitations, and possible applications of the framework in nonequilibrium statistical mechanics. In particular, we emphasize that the method is intended as a systematic observable-based formulation for systems governed by linear evolution equations, rather than as a universal closure scheme for arbitrary nonequilibrium dynamics.<\/jats:p>","DOI":"10.3390\/e28060654","type":"journal-article","created":{"date-parts":[[2026,6,11]],"date-time":"2026-06-11T05:50:58Z","timestamp":1781157058000},"page":"654","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Expectation Identities for Dynamical Systems: A Classical Analog of the Ehrenfest Theorem"],"prefix":"10.3390","volume":"28","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7633-2739","authenticated-orcid":false,"given":"Abiam","family":"Tamburrini","sequence":"first","affiliation":[{"name":"Dipartimento di Fisica, Universit\u00e1 Della Calabria, I-87036 Rende, Italy"},{"name":"Departamento de F\u00edsica, Facultad de Ciencias, Universidad de Chile, Santiago 8370459, Chile"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2757-332X","authenticated-orcid":false,"given":"Sergio","family":"Davis","sequence":"additional","affiliation":[{"name":"Departamento de F\u00edsica y Astronom\u00eda, Facultad de Ciencias Exactas, Universidad Andres Bello, Sazi\u00e9 2212, Piso 7, Santiago 8370136, Chile"},{"name":"Research Center in the Intersection of Plasma Physics, Matter and Complexity (P<sup>2<\/sup>mc), Comisi\u00f3n Chilena de Energ\u00eda Nuclear, Casilla 188-D, Santiago 8340701, Chile"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Diego","family":"Gonz\u00e1lez","sequence":"additional","affiliation":[{"name":"Departamento de F\u00edsica, Facultad de Ciencias, Universidad Cat\u00f3lica del Norte, Av. Angamos 0610, Antofagasta 1270398, Chile"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9161-0888","authenticated-orcid":false,"given":"Pablo S.","family":"Moya","sequence":"additional","affiliation":[{"name":"Departamento de F\u00edsica, Facultad de Ciencias, Universidad de Chile, Santiago 8370459, Chile"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2026,6,9]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Risken, H. (1996). The Fokker-Planck Equation: Methods of Solution and Applications, Springer.","DOI":"10.1007\/978-3-642-61544-3"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Zwanzig, R. (2001). Nonequilibrium Statistical Mechanics, Oxford University Press.","DOI":"10.1093\/oso\/9780195140187.001.0001"},{"key":"ref_3","unstructured":"Greiner, W., Neise, L., and St\u00f6cker, H. (2012). 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