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Rev. Lett."],"accepted":{"date-parts":[[2025,6,10]]},"abstract":"<jats:p>It is a long-standing challenge to accurately and efficiently compute thermodynamic quantities of many-body systems at thermal equilibrium. The conventional methods, e.g., Markov chain Monte\u00a0Carlo, require many steps to equilibrate. The recently developed deep learning methods can perform direct sampling, but only work at a single trained temperature point and risk biased sampling. Here, we propose a variational method for canonical ensembles with differentiable temperatures, which gives thermodynamic quantities as continuous functions of temperature akin to an analytical solution. The proposed method is a general framework that works with  tractable density generative model. At optimal, the model is theoretically guaranteed to be the  Boltzmann distribution. We validated our method by calculating phase transitions in the Ising and XY models, demonstrating that our direct-sampling simulations are as accurate as Markov chain Monte\u00a0Carlo methods but more efficient. Moreover, our differentiable free energy aligns closely with the exact one to the second-order derivative, indicating that the variational model captures the subtle thermal transitions at the phase transitions. This functional dependence on external parameters is a fundamental advancement in combining the exceptional fitting ability of deep learning with rigorous physical analysis.<\/jats:p>","DOI":"10.1103\/8wx7-kyx8","type":"journal-article","created":{"date-parts":[[2025,6,10]],"date-time":"2025-06-10T07:48:47Z","timestamp":1749541727000},"update-policy":"https:\/\/doi.org\/10.1103\/crossmark-policy","source":"Crossref","is-referenced-by-count":2,"title":["Deep Generative Modeling of the Canonical Ensemble with Differentiable Thermal Properties"],"prefix":"10.1103","volume":"135","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4262-5768","authenticated-orcid":true,"given":"Shuo-Hui","family":"Li","sequence":"first","affiliation":[{"id":[{"id":"https:\/\/ror.org\/00q4vv597","id-type":"ROR","asserted-by":"publisher"}],"name":"The Hong Kong University of Science and Technology"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0008-6664-9706","authenticated-orcid":true,"given":"Yao-Wen","family":"Zhang","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/00q4vv597","id-type":"ROR","asserted-by":"publisher"}],"name":"The Hong Kong University of Science and Technology"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2353-0130","authenticated-orcid":true,"given":"Ding","family":"Pan","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/00q4vv597","id-type":"ROR","asserted-by":"publisher"}],"name":"The Hong Kong University of Science and Technology"},{"id":[{"id":"https:\/\/ror.org\/00q4vv597","id-type":"ROR","asserted-by":"publisher"}],"name":"The Hong Kong University of Science and Technology"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"16","published-online":{"date-parts":[[2025,7,8]]},"reference":[{"key":"8wx7-kyx8Cc1R1","doi-asserted-by":"publisher","DOI":"10.1016\/j.tcs.2005.09.011","type":"journal-article"},{"key":"8wx7-kyx8Cc2R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.99.120601","type":"journal-article"},{"key":"8wx7-kyx8Cc3R1","doi-asserted-by":"publisher","DOI":"10.1023\/A:1007665907178","type":"journal-article"},{"key":"8wx7-kyx8Cc4R1","volume-title":"Statistical Mechanics","author":"R.\u2009K. 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All rights are reserved, including those for text and data mining, AI training, and similar technologies.","name":"copyright_statement","label":"Copyright statement","group":{"name":"copyright","label":"copyright"}},{"value":"2025","name":"copyright_year","label":"Copyright year","group":{"name":"copyright","label":"copyright"}},{"value":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/","name":"creative_commons_license","label":"Creative Commons license","group":{"name":"licenses","label":"Licenses"}},{"value":"\u00a92025 The authors","name":"copyright_holder","label":"Copyright holder","group":{"name":"copyright","label":"copyright"}}],"article-number":"027301"}}