{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,21]],"date-time":"2026-01-21T12:32:42Z","timestamp":1768998762886,"version":"3.49.0"},"reference-count":22,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2020,8,10]],"date-time":"2020-08-10T00:00:00Z","timestamp":1597017600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100003407","name":"Ministero dell\u2019Istruzione, dell\u2019Universit\u00e0 e della Ricerca","doi-asserted-by":"publisher","award":["Dipartimenti di Ecellenza 2018\u20132022"],"award-info":[{"award-number":["Dipartimenti di Ecellenza 2018\u20132022"]}],"id":[{"id":"10.13039\/501100003407","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Time reversal invariance (TRI) of particles systems has many consequences, among which the celebrated Onsager reciprocal relations, a milestone in Statistical Mechanics dating back to 1931. Because for a long time it was believed that (TRI) dos not hold in presence of a magnetic field, a modification of such relations was proposed by Casimir in 1945. Only in the last decade, the strict traditional notion of reversibility that led to Casimir\u2019s work has been questioned. It was then found that other symmetries can be used, which allow the Onsager reciprocal relations to hold without modification. In this paper we advance this investigation for classical Hamiltonian systems, substantially increasing the number of symmetries that yield TRI in presence of a magnetic field. We first deduce the most general form of a generalized time reversal operation on the phase space of such a system; secondly, we express sufficient conditions on the magnetic field which ensure TRI. Finally, we examine common examples from statistical mechanics and molecular dynamics. Our main result is that TRI holds in a much wider generality than previously believed, partially explaining why no experimental violation of Onsager relations has so far been reported.<\/jats:p>","DOI":"10.3390\/sym12081336","type":"journal-article","created":{"date-parts":[[2020,8,10]],"date-time":"2020-08-10T09:04:16Z","timestamp":1597050256000},"page":"1336","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Necessary and Sufficient Conditions for Time Reversal Symmetry in Presence of Magnetic Fields"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2859-6603","authenticated-orcid":false,"given":"Davide","family":"Carbone","sequence":"first","affiliation":[{"name":"Dipartimento di Fisica, Universit\u00e0 di Torino, Via Pietro Giuria 1, 10125 Torino, 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"}]}],"member":"1968","published-online":{"date-parts":[[2020,8,10]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"405","DOI":"10.1103\/PhysRev.37.405","article-title":"Reciprocal relations in irreversible processes. 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Matter Complex Syst."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/8\/1336\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:58:46Z","timestamp":1760176726000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/8\/1336"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,8,10]]},"references-count":22,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2020,8]]}},"alternative-id":["sym12081336"],"URL":"https:\/\/doi.org\/10.3390\/sym12081336","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,8,10]]}}}