{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,10]],"date-time":"2026-03-10T01:04:38Z","timestamp":1773104678731,"version":"3.50.1"},"reference-count":31,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2024,4,26]],"date-time":"2024-04-26T00:00:00Z","timestamp":1714089600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"European Union-NextGenerationEU within the National Center for HPC, Big Data and Quantum Computing","award":["CN00000013"],"award-info":[{"award-number":["CN00000013"]}]},{"name":"University of Padova","award":["CN00000013"],"award-info":[{"award-number":["CN00000013"]}]},{"name":"Iniziativa Specifica Quantum of Istituto Nazionale di Fisica Nucleare","award":["CN00000013"],"award-info":[{"award-number":["CN00000013"]}]},{"name":"European Quantum Flagship Project PASQuanS 2","award":["CN00000013"],"award-info":[{"award-number":["CN00000013"]}]},{"name":"RIN 2022 Project Quantum Atomic Mixtures: Droplets, Topological Structures, and Vortices","award":["CN00000013"],"award-info":[{"award-number":["CN00000013"]}]},{"name":"Project Frontiere Quantistiche within the 2023 funding programme \u2019Dipartimenti di Eccellenza\u2019 of the Italian Ministry for Universities and Research","award":["CN00000013"],"award-info":[{"award-number":["CN00000013"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We analyze the general relation between canonical and grand canonical ensembles in the thermodynamic limit. We begin our discussion by deriving, with an alternative approach, some standard results first obtained by Kac and coworkers in the late 1970s. Then, motivated by the Bose\u2013Einstein condensation (BEC) of trapped gases with a fixed number of atoms, which is well described by the canonical ensemble and by the recent groundbreaking experimental realization of BEC with photons in a dye-filled optical microcavity under genuine grand canonical conditions, we apply our formalism to a system of non-interacting Bose particles confined in a two-dimensional harmonic trap. We discuss in detail the mathematical origin of the inequivalence of ensembles observed in the condensed phase, giving place to the so-called grand canonical catastrophe of density fluctuations. We also provide explicit analytical expressions for the internal energy and specific heat and compare them with available experimental data. For these quantities, we show the equivalence of ensembles in the thermodynamic limit.<\/jats:p>","DOI":"10.3390\/e26050367","type":"journal-article","created":{"date-parts":[[2024,4,29]],"date-time":"2024-04-29T10:33:36Z","timestamp":1714386816000},"page":"367","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Canonical vs. Grand Canonical Ensemble for Bosonic Gases under Harmonic Confinement"],"prefix":"10.3390","volume":"26","author":[{"given":"Andrea","family":"Crisanti","sequence":"first","affiliation":[{"name":"Dipartimento di Fisica, Universit\u00e0 di Roma \u201cLa Sapienza\u201d, Piazzale Moro 5, 00185 Roma, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0817-4753","authenticated-orcid":false,"given":"Luca","family":"Salasnich","sequence":"additional","affiliation":[{"name":"Dipartimento di Fisica e Astronomia \u201cGalileo Galilei\u201d and Padua QTech Center, Universit\u00e0 di Padova, Via Marzolo 8, 35131 Padova, Italy"},{"name":"Istituto Nazionale di Fisica Nucleare, Sezione di Padova, Via Marzolo 8, 35131 Padova, Italy"},{"name":"Istituto Nazionale di Ottica del Consiglio Nazionale Delle Ricerche, Via Nello Carrara 2, 50127 Sesto Fiorentino, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3686-0822","authenticated-orcid":false,"given":"Alessandro","family":"Sarracino","sequence":"additional","affiliation":[{"name":"Dipartimento di Ingegneria, Universit\u00e0 della Campania \u201cLuigi Vanvitelli\u201d, Via Roma 29, 81031 Aversa, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Marco","family":"Zannetti","sequence":"additional","affiliation":[{"name":"Dipartimento di Fisica \u201cEduardo Caianiello\u201d, Universit\u00e0 di Salerno, Via Giovanni Paolo II 132, 84084 Salerno, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,4,26]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"198","DOI":"10.1126\/science.269.5221.198","article-title":"Observation of Bose-Einstein condensation in a dilute atomic vapor","volume":"269","author":"Anderson","year":"1995","journal-title":"Science"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"3969","DOI":"10.1103\/PhysRevLett.75.3969","article-title":"Bose-Einstein condensation in a gas of sodium atoms","volume":"75","author":"Davis","year":"1995","journal-title":"Phys. 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