{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,28]],"date-time":"2026-04-28T05:23:29Z","timestamp":1777353809248,"version":"3.51.4"},"reference-count":38,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2021,10,27]],"date-time":"2021-10-27T00:00:00Z","timestamp":1635292800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100007587","name":"Aix-Marseille University","doi-asserted-by":"publisher","award":["MOLINT"],"award-info":[{"award-number":["MOLINT"]}],"id":[{"id":"10.13039\/100007587","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Different arguments led to supposing that the deep origin of phase transitions has to be identified with suitable topological changes of potential related submanifolds of configuration space of a physical system. An important step forward for this approach was achieved with two theorems stating that, for a wide class of physical systems, phase transitions should necessarily stem from topological changes of energy level submanifolds of the phase space. However, the sufficiency conditions are still a wide open question. In this study, a first important step forward was performed in this direction; in fact, a differential equation was worked out which describes how entropy varies as a function of total energy, and this variation is driven by the total energy dependence of a topology-related quantity of the relevant submanifolds of the phase space. Hence, general conditions can be in principle defined for topology-driven loss of differentiability of the entropy.<\/jats:p>","DOI":"10.3390\/e23111414","type":"journal-article","created":{"date-parts":[[2021,10,27]],"date-time":"2021-10-27T22:00:23Z","timestamp":1635372023000},"page":"1414","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Topology and Phase Transitions: A First Analytical Step towards the Definition of Sufficient Conditions"],"prefix":"10.3390","volume":"23","author":[{"given":"Loris","family":"Di Cairano","sequence":"first","affiliation":[{"name":"Institute of Neuroscience and Medicine INM-9, and Institute for Advanced Simulation IAS-5, Forschungszentrum J\u00fclich, 52428 J\u00fclich, Germany"},{"name":"Department of Physics, Faculty of Mathematics, Computer Science and Natural Sciences, Aachen University, 52062 Aachen, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Matteo","family":"Gori","sequence":"additional","affiliation":[{"name":"Physics and Materials Science Research Unit, University of Luxembourg, L-1511 Luxembourg, Luxembourg"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Marco","family":"Pettini","sequence":"additional","affiliation":[{"name":"Aix-Marseille Univ, CNRS, Universit\u00e9 de Toulon, 13288 Marseille, France"},{"name":"CNRS Centre de Physique Th\u00e9orique UMR7332, 13288 Marseille, France"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,10,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"828","DOI":"10.1103\/PhysRevE.47.828","article-title":"Geometrical hints for a nonperturbative approach to Hamiltonian dynamics","volume":"47","author":"Pettini","year":"1993","journal-title":"Phys. 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