{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,20]],"date-time":"2025-12-20T22:27:33Z","timestamp":1766269653213,"version":"build-2065373602"},"reference-count":143,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2018,10,16]],"date-time":"2018-10-16T00:00:00Z","timestamp":1539648000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Many open quantum systems encountered in both natural and synthetic situations are embedded in classical-like baths. Often, the bath degrees of freedom may be represented in terms of canonically conjugate coordinates, but in some cases they may require a non-canonical or non-Hamiltonian representation. Herein, we review an approach to the dynamics and statistical mechanics of quantum subsystems embedded in either non-canonical or non-Hamiltonian classical-like baths which is based on operator-valued quasi-probability functions. These functions typically evolve through the action of quasi-Lie brackets and their associated Quantum-Classical Liouville Equations, or through quasi-Lie brackets augmented by dissipative terms. Quasi-Lie brackets possess the unique feature that, while conserving the energy (which the Noether theorem links to time-translation symmetry), they violate the time-translation symmetry of their algebra. This fact can be heuristically understood in terms of the dynamics of the open quantum subsystem. We then describe an example in which a quantum subsystem is embedded in a bath of classical spins, which are described by non-canonical coordinates. In this case, it has been shown that an off-diagonal open-bath geometric phase enters into the propagation of the quantum-classical dynamics. Next, we discuss how non-Hamiltonian dynamics may be employed to generate the constant-temperature evolution of phase space degrees of freedom coupled to the quantum subsystem. Constant-temperature dynamics may be generated by either a classical Langevin stochastic process or a Nos\u00e9\u2013Hoover deterministic thermostat. These two approaches are not equivalent but have different advantages and drawbacks. In all cases, the calculation of the operator-valued quasi-probability function allows one to compute time-dependent statistical averages of observables. This may be accomplished in practice using a hybrid Molecular Dynamics\/Monte Carlo algorithms, which we outline herein.<\/jats:p>","DOI":"10.3390\/sym10100518","type":"journal-article","created":{"date-parts":[[2018,10,16]],"date-time":"2018-10-16T11:07:51Z","timestamp":1539688071000},"page":"518","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":18,"title":["Quasi-Lie Brackets and the Breaking of Time-Translation Symmetry for Quantum Systems Embedded in Classical Baths"],"prefix":"10.3390","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3954-0445","authenticated-orcid":false,"given":"Alessandro","family":"Sergi","sequence":"first","affiliation":[{"name":"Dipartimento di Scienze Matematiche e Informatiche, Scienze Fisiche e Scienze della Terra, Universit\u00e0 degli Studi di Messina, Contrada Papardo, 98166 Messina, Italy"},{"name":"Institute of Systems Science, Durban University of Technology, P.O. Box 1334, Durban 4000, South Africa"},{"name":"Istituto Nazionale di Fisica Nucleare, Sez. di Catania,95123 Catania, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Gabriel","family":"Hanna","sequence":"additional","affiliation":[{"name":"Department of Chemistry, University of Alberta, 11227 Saskatchewan Drive Edmonton, Edmonton, AB T6G 2G2, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Roberto","family":"Grimaudo","sequence":"additional","affiliation":[{"name":"Istituto Nazionale di Fisica Nucleare, Sez. di Catania,95123 Catania, Italy"},{"name":"Dipartimento di Fisica e Chimica dell\u2019Universit\u00e1 di Palermo, Via Archirafi 36, I-90123 Palermo, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Antonino","family":"Messina","sequence":"additional","affiliation":[{"name":"Istituto Nazionale di Fisica Nucleare, Sez. di Catania,95123 Catania, Italy"},{"name":"Dipartimento di Matematica e Informatica, Universit\u00e0 degli Studi di Palermo, Via Archirafi 34, I-90123 Palermo, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,10,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Caldeira, A.O. 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