{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T13:27:43Z","timestamp":1753882063233,"version":"3.41.2"},"reference-count":9,"publisher":"World Scientific Pub Co Pte Ltd","issue":"04","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Open Syst. Inf. Dyn."],"published-print":{"date-parts":[[2023,12]]},"abstract":"<jats:p> Let [Formula: see text] denote the set of [Formula: see text] by [Formula: see text] complex matrices. Consider continuous time quantum semigroups [Formula: see text], [Formula: see text], where [Formula: see text] is the infinitesimal generator. If we assume that [Formula: see text], we will call [Formula: see text], [Formula: see text] a quantum Markov semigroup. Given a stationary density matrix [Formula: see text], for the quantum Markov semigroup [Formula: see text], [Formula: see text], we can define a continuous time stationary quantum Markov process, denoted by [Formula: see text], [Formula: see text] Given an a priori Laplacian operator [Formula: see text], we will present a natural concept of entropy for a class of density matrices on [Formula: see text]. Given a Hermitian operator [Formula: see text] (which plays the role of a Hamiltonian), we will study a version of the variational principle of pressure for [Formula: see text]. A density matrix [Formula: see text] maximizing pressure will be called an equilibrium density matrix. From [Formula: see text] we will derive a new infinitesimal generator [Formula: see text]. Finally, the continuous time quantum Markov process defined by the semigroup [Formula: see text], [Formula: see text], and an initial stationary density matrix, will be called the continuous time equilibrium quantum Markov process for the Hamiltonian [Formula: see text]. It corresponds to the quantum thermodynamical equilibrium for the action of the Hamiltonian [Formula: see text]. <\/jats:p>","DOI":"10.1142\/s123016122350018x","type":"journal-article","created":{"date-parts":[[2024,2,2]],"date-time":"2024-02-02T09:01:59Z","timestamp":1706864519000},"source":"Crossref","is-referenced-by-count":1,"title":["Thermodynamic Formalism for Continuous-Time Quantum Markov Semigroups: the Detailed Balance Condition, Entropy, Pressure and Equilibrium Quantum Processes"],"prefix":"10.1142","volume":"30","author":[{"given":"Jader E.","family":"Brasil","sequence":"first","affiliation":[{"name":"IME\u2013UFRGS, Porto Alegre, Brazil"}]},{"given":"Josu\u00e9","family":"Knorst","sequence":"additional","affiliation":[{"name":"IME\u2013UFRGS, Porto Alegre, Brazil"}]},{"given":"Artur O.","family":"Lopes","sequence":"additional","affiliation":[{"name":"IME\u2013UFRGS, Porto Alegre, Brazil"}]}],"member":"219","published-online":{"date-parts":[[2024,1,23]]},"reference":[{"key":"S123016122350018XBIB001","doi-asserted-by":"publisher","DOI":"10.11606\/issn.2316-9028.v4i1p1-16"},{"key":"S123016122350018XBIB003","doi-asserted-by":"publisher","DOI":"10.1016\/j.jfa.2017.05.003"},{"key":"S123016122350018XBIB004","doi-asserted-by":"publisher","DOI":"10.1142\/S0219025707002762"},{"key":"S123016122350018XBIB005","doi-asserted-by":"publisher","DOI":"10.1007\/s00220-010-1011-1"},{"volume-title":"Acad. Naz. dei Lincei Scuola Normale Superiore","year":"1980","author":"Kac M.","key":"S123016122350018XBIB006"},{"key":"S123016122350018XBIB008","doi-asserted-by":"publisher","DOI":"10.1007\/s10955-013-0796-7"},{"key":"S123016122350018XBIB009","doi-asserted-by":"publisher","DOI":"10.1017\/etds.2014.15"},{"key":"S123016122350018XBIB010","first-page":"1","volume":"187","author":"Parry W.","year":"1990","journal-title":"Asterisque"},{"key":"S123016122350018XBIB011","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4613-8514-1"}],"container-title":["Open Systems &amp; Information Dynamics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S123016122350018X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,2,2]],"date-time":"2024-02-02T09:02:18Z","timestamp":1706864538000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/10.1142\/S123016122350018X"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,12]]},"references-count":9,"journal-issue":{"issue":"04","published-print":{"date-parts":[[2023,12]]}},"alternative-id":["10.1142\/S123016122350018X"],"URL":"https:\/\/doi.org\/10.1142\/s123016122350018x","relation":{},"ISSN":["1230-1612","1793-7191"],"issn-type":[{"type":"print","value":"1230-1612"},{"type":"electronic","value":"1793-7191"}],"subject":[],"published":{"date-parts":[[2023,12]]},"article-number":"2350018"}}