{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:37:05Z","timestamp":1760060225310,"version":"build-2065373602"},"reference-count":54,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2025,8,11]],"date-time":"2025-08-11T00:00:00Z","timestamp":1754870400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Italian National Group of Mathematical Physics"},{"name":"European Union\u2014NextGenerationEU"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>We develop a general framework for response theory in Markovian diffusion processes governed by Fokker\u2013Planck equations. Our formalism, based on the notion of the dissipation function, is capable of handling dynamics that are driven by an external perturbation arbitrarily far from a given reference process. Using the analytically solvable Brownian oscillator model, we derive exact response formulae for both overdamped and underdamped dynamics of harmonically bound Brownian particles. We also demonstrate that for certain observables and under suitable time scaling, the operations of model reduction and response formula extraction commute, which highlights a relevant symmetry of the adopted mathematical formalism. Notably, the time reversal invariance symmetry, which is manifested as detailed balance in stochastic processes and is often required in statistical mechanics, is not necessary in our response framework.<\/jats:p>","DOI":"10.3390\/sym17081297","type":"journal-article","created":{"date-parts":[[2025,8,11]],"date-time":"2025-08-11T14:32:36Z","timestamp":1754922756000},"page":"1297","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Dissipation Functions and Brownian Oscillators"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7424-7888","authenticated-orcid":false,"given":"Matteo","family":"Colangeli","sequence":"first","affiliation":[{"name":"Dipartimento di Ingegneria e Scienze dell\u2019Informazione e Matematica, Universit\u00e0 degli Studi dell\u2019Aquila, Via Vetoio, 67100 L\u2019Aquila, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4223-6279","authenticated-orcid":false,"given":"Lamberto","family":"Rondoni","sequence":"additional","affiliation":[{"name":"Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy"},{"name":"Istituto Nazionale di Fisica Nucleare, Sezione di Torino, Via Pietro Giuria 1, 10125 Torino, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0000-7609-1947","authenticated-orcid":false,"given":"Pasquale","family":"Vozza","sequence":"additional","affiliation":[{"name":"Dipartimento di Fisica, Universit\u00e0 degli Studi di Torino, Via Pietro Giuria 1, 10125 Torino, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,8,11]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1088\/0034-4885\/29\/1\/306","article-title":"The Fluctuation-Dissipation Theorem","volume":"29","author":"Kubo","year":"1966","journal-title":"Rep. 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