{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,10]],"date-time":"2026-03-10T22:47:38Z","timestamp":1773182858223,"version":"3.50.1"},"reference-count":41,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2024,2,17]],"date-time":"2024-02-17T00:00:00Z","timestamp":1708128000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Italian Ministry of University and Research (MUR)","award":["P2022Z7ZAJ"],"award-info":[{"award-number":["P2022Z7ZAJ"]}]},{"name":"Italian National Group of Mathematical Physics (GNFM)","award":["P2022Z7ZAJ"],"award-info":[{"award-number":["P2022Z7ZAJ"]}]},{"name":"Istituto Nazionale di Alta Matematica (INdAM)","award":["P2022Z7ZAJ"],"award-info":[{"award-number":["P2022Z7ZAJ"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We review, under a modern light, the conditions that render the Boltzmann equation applicable. These are conditions that permit probability to behave like mass, thereby possessing clear and concrete content, whereas generally, this is not the case. Because science and technology are increasingly interested in small systems that violate the conditions of the Boltzmann equation, probability appears to be the only mathematical tool suitable for treating them. Therefore, Boltzmann\u2019s teachings remain relevant, and the present analysis provides a critical perspective useful for accurately interpreting the results of current applications of statistical mechanics.<\/jats:p>","DOI":"10.3390\/e26020171","type":"journal-article","created":{"date-parts":[[2024,2,19]],"date-time":"2024-02-19T04:39:36Z","timestamp":1708317576000},"page":"171","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Probability Turns Material: The Boltzmann Equation"],"prefix":"10.3390","volume":"26","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4223-6279","authenticated-orcid":false,"given":"Lamberto","family":"Rondoni","sequence":"first","affiliation":[{"name":"Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca Degli Abruzzi 24, 10129 Turin, Italy"},{"name":"INFN, Sezione di Torino, Via Pietro Giuria 1, 10125 Turin, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9839-1518","authenticated-orcid":false,"given":"Vincenzo","family":"Di Florio","sequence":"additional","affiliation":[{"name":"Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca Degli Abruzzi 24, 10129 Turin, Italy"},{"name":"INFN, Sezione di Torino, Via Pietro Giuria 1, 10125 Turin, Italy"},{"name":"Istituto Italiano di Tecnologia (IIT), Via Melen 83, B Block, 16163 Genoa, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,2,17]]},"reference":[{"key":"ref_1","unstructured":"Prigogine, I. 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