{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:29:31Z","timestamp":1760146171340,"version":"build-2065373602"},"reference-count":82,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2024,10,5]],"date-time":"2024-10-05T00:00:00Z","timestamp":1728086400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"European Union\u2019s Horizon 2020 Research and Innovation Programme","award":["964203"],"award-info":[{"award-number":["964203"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this review work, we outline a conceptual path that, starting from the numerical investigation of the transition between weak chaos and strong chaos in Hamiltonian systems with many degrees of freedom, comes to highlight how, at the basis of equilibrium phase transitions, there must be major changes in the topology of submanifolds of the phase space of Hamiltonian systems that describe systems that exhibit phase transitions. In fact, the numerical investigation of Hamiltonian flows of a large number of degrees of freedom that undergo a thermodynamic phase transition has revealed peculiar dynamical signatures detected through the energy dependence of the largest Lyapunov exponent, that is, of the degree of chaoticity of the dynamics at the phase transition point. The geometrization of Hamiltonian flows in terms of geodesic flows on suitably defined Riemannian manifolds, used to explain the origin of deterministic chaos, combined with the investigation of the dynamical counterpart of phase transitions unveils peculiar geometrical changes of the mechanical manifolds in correspondence to the peculiar dynamical changes at the phase transition point. Then, it turns out that these peculiar geometrical changes are the effect of deeper topological changes of the configuration space hypersurfaces\u00a0\u2211v=VN\u22121(v)\u00a0as well as of the manifolds\u00a0{Mv=VN\u22121((\u2212\u221e,v])}v\u2208R\u00a0bounded by the \u2211v. In other words, denoting by vc the critical value of the average potential energy density at which the phase transition takes place, the members of the family\u00a0{\u2211v}v&lt;vc\u00a0are not diffeomorphic to those of the family\u00a0{\u2211v}v&gt;vc; additionally, the members of the family\u00a0{Mv}v&gt;vc\u00a0are not diffeomorphic to those of\u00a0{Mv}v&gt;vc. The topological theory of the deep origin of phase transitions allows a unifying framework to tackle phase transitions that may or may not be due to a symmetry-breaking phenomenon (that is, with or without an order parameter) and to finite\/small N systems.<\/jats:p>","DOI":"10.3390\/e26100840","type":"journal-article","created":{"date-parts":[[2024,10,7]],"date-time":"2024-10-07T08:35:23Z","timestamp":1728290123000},"page":"840","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["From Geometry of Hamiltonian Dynamics to Topology of Phase Transitions: A Review"],"prefix":"10.3390","volume":"26","author":[{"given":"Giulio","family":"Pettini","sequence":"first","affiliation":[{"name":"Dipartimento di Fisica, Universit\u00e0 di Firenze, and I.N.F.N., Sezione di Firenze, via G. Sansone 1, I-50019 Sesto Fiorentino, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Matteo","family":"Gori","sequence":"additional","affiliation":[{"name":"Department of Physics and Sciences of Materials, 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":"Centre de Physique Th\u00e9orique, 13288 Marseille, France"},{"name":"Quantum Biology Lab, Howard University, Washington, DC 20059, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,10,5]]},"reference":[{"key":"ref_1","first-page":"926","article-title":"On the Theory of Phase Transitions","volume":"7","author":"Landau","year":"1937","journal-title":"Zh. Eksp. Teor. 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