{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T00:58:54Z","timestamp":1760057934091,"version":"build-2065373602"},"reference-count":49,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2025,2,28]],"date-time":"2025-02-28T00:00:00Z","timestamp":1740700800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"RESEARCH SUPPORT PLAN 2022-Call"},{"name":"Italian National Group of Mathematical Physics (GNFM-INdAM)"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Resorting to microcanonical ensemble Monte Carlo simulations, we study the geometric and topological properties of the state space of a model of a network glass-former. This model, a Lennard-Jones binary mixture, does not crystallize due to frustration. We have found two peaks in specific heat at equilibrium and at low energy, corresponding to important changes in local ordering. These singularities were accompanied by inflection points in geometrical markers of the potential energy level sets\u2014namely, the mean curvature, the dispersion of the principal curvatures, and the variance of the scalar curvature. Pinkall\u2019s and Overholt\u2019s theorems closely relate these quantities to the topological properties of the accessible state-space manifold. Thus, our analysis provides strong indications that the glass transition is associated with major changes in the topology of the energy level sets. This important result suggests that this phase transition can be understood through the topological theory of phase transitions.<\/jats:p>","DOI":"10.3390\/e27030258","type":"journal-article","created":{"date-parts":[[2025,2,28]],"date-time":"2025-02-28T10:12:33Z","timestamp":1740737553000},"page":"258","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["The Glass Transition: A Topological Perspective"],"prefix":"10.3390","volume":"27","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1823-4406","authenticated-orcid":false,"given":"Arthur","family":"Vesperini","sequence":"first","affiliation":[{"name":"Department of Physical Sciences, Earth and Environment (DSFTA), University of Siena, Via Roma 56, 53100 Siena, Italy"},{"name":"INFN Sezione di Perugia, 06123 Perugia, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7588-921X","authenticated-orcid":false,"given":"Roberto","family":"Franzosi","sequence":"additional","affiliation":[{"name":"Department of Physical Sciences, Earth and Environment (DSFTA), University of Siena, Via Roma 56, 53100 Siena, Italy"},{"name":"INFN Sezione di Perugia, 06123 Perugia, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1561-4390","authenticated-orcid":false,"given":"Marco","family":"Pettini","sequence":"additional","affiliation":[{"name":"Aix-Marseille University, Universit\u00e9 de Toulon, CNRS, 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":[[2025,2,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Pettini, M. (2007). Geometry and Topology in Hamiltonian Dynamics and Statistical Mechanics, Springer. Interdisciplinary Applied Mathematics.","DOI":"10.1007\/978-0-387-49957-4"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"863","DOI":"10.1016\/0022-3697(88)90002-9","article-title":"Perspective on the Glass Transition","volume":"49","author":"Angell","year":"1988","journal-title":"J. Phys. Chem. Solids"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1016\/S0378-4371(99)00626-3","article-title":"The Physics of the Glass Transition","volume":"280","author":"Parisi","year":"2000","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"5356","DOI":"10.1103\/PhysRevLett.85.5356","article-title":"Saddles in the Energy Landscape Probed by Supercooled Liquids","volume":"85","author":"Angelani","year":"2000","journal-title":"Phys. Rev. Lett."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"5360","DOI":"10.1103\/PhysRevLett.85.5360","article-title":"Energy Landscape of a Lennard-Jones Liquid: Statistics of Stationary Points","volume":"85","author":"Broderix","year":"2000","journal-title":"Phys. Rev. Lett."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"064502","DOI":"10.1063\/1.2151899","article-title":"Geometrical Properties of the Potential Energy of the Soft-Sphere Binary Mixture","volume":"124","author":"Grigera","year":"2006","journal-title":"J. Chem. Phys."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"055502","DOI":"10.1103\/PhysRevLett.88.055502","article-title":"Geometric Approach to the Dynamic Glass Transition","volume":"88","author":"Grigera","year":"2002","journal-title":"Phys. Rev. Lett."},{"key":"ref_8","unstructured":"Morse, M. (2014). The Calculus of Variations in the Large, repr ed., American Mathematical Society. Number 18 in Colloquium Publications\/American Mathematical Society."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"093204","DOI":"10.1088\/1742-5468\/aad6b6","article-title":"Topological Origin of Phase Transitions in the Absence of Critical Points of the Energy Landscape","volume":"2018","author":"Gori","year":"2018","journal-title":"J. Stat. Mech. Theory Exp."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Di Cairano, L., Gori, M., and Pettini, M. (2021). Topology and Phase Transitions: A First Analytical Step towards the Definition of Sufficient Conditions. Entropy, 23.","DOI":"10.3390\/e23111414"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"132909","DOI":"10.1016\/j.physd.2021.132909","article-title":"Hamiltonian Chaos and Differential Geometry of Configuration Space\u2013Time","volume":"422","author":"Gori","year":"2021","journal-title":"Phys. D Nonlinear Phenom."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"3182","DOI":"10.1063\/1.1732447","article-title":"Thermodynamics of Small Systems","volume":"36","author":"Hill","year":"1962","journal-title":"J. Chem. Phys."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"224111","DOI":"10.1063\/1.1901658","article-title":"The Microcanonical Thermodynamics of Finite Systems: The Microscopic Origin of Condensation and Phase Separations, and the Conditions for Heat Flow from Lower to Higher Temperatures","volume":"122","author":"Gross","year":"2005","journal-title":"J. Chem. Phys."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"974","DOI":"10.1038\/ncomms1974","article-title":"Roles of Icosahedral and Crystal-like Order in the Hard Spheres Glass Transition","volume":"3","author":"Leocmach","year":"2012","journal-title":"Nat. Commun."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1140\/epje\/i2012-12113-y","article-title":"Bond Orientational Order in Liquids: Towards a Unified Description of Water-like Anomalies, Liquid-Liquid Transition, Glass Transition, and Crystallization: Bond Orientational Order in Liquids","volume":"35","author":"Tanaka","year":"2012","journal-title":"Eur. Phys. J. E"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"064201","DOI":"10.1103\/PhysRevB.70.064201","article-title":"Local Properties of the Potential-Energy Landscape of a Model Glass: Understanding the Low-Temperature Anomalies","volume":"70","author":"Reinisch","year":"2004","journal-title":"Phys. Rev. B"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"9039","DOI":"10.1063\/1.480246","article-title":"Thermodynamics of Binary Mixture Glasses","volume":"111","author":"Coluzzi","year":"1999","journal-title":"J. Chem. Phys."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"124504","DOI":"10.1063\/1.2773716","article-title":"Understanding Fragility in Supercooled Lennard-Jones Mixtures. I. Locally Preferred Structures","volume":"127","author":"Coslovich","year":"2007","journal-title":"J. Chem. Phys."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"4891","DOI":"10.1103\/PhysRevA.36.4891","article-title":"Soft-Sphere Model for the Glass Transition in Binary Alloys: Pair Structure and Self-Diffusion","volume":"36","author":"Bernu","year":"1987","journal-title":"Phys. Rev. A"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"4626","DOI":"10.1103\/PhysRevE.51.4626","article-title":"Testing Mode-Coupling Theory for a Supercooled Binary Lennard-Jones Mixture I: The van Hove Correlation Function","volume":"51","author":"Kob","year":"1995","journal-title":"Phys. Rev. E"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"061505","DOI":"10.1103\/PhysRevE.73.061505","article-title":"Hybrid Monte Carlo Simulation of a Glass-Forming Binary Mixture","volume":"73","author":"Flenner","year":"2006","journal-title":"Phys. Rev. E"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Chamberlin, R.V. (2022). An Ising Model for Supercooled Liquids and the Glass Transition. Symmetry, 14.","DOI":"10.3390\/sym14102211"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Bel-Hadj-Aissa, G., Gori, M., Penna, V., Pettini, G., and Franzosi, R. (2020). Geometrical Aspects in the Analysis of Microcanonical Phase-Transitions. Entropy, 22.","DOI":"10.3390\/e22040380"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"3030","DOI":"10.1103\/PhysRevA.32.3030","article-title":"Laplace-Transform Technique for Deriving Thermodynamic Equations from the Classical Microcanonical Ensemble","volume":"32","author":"Pearson","year":"1985","journal-title":"Phys. Rev. A"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"054134","DOI":"10.1103\/PhysRevE.106.054134","article-title":"Topological Origin of the Protein Folding Transition","volume":"106","author":"Capelli","year":"2022","journal-title":"Phys. Rev. E"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"023206","DOI":"10.1088\/1742-5468\/abda27","article-title":"Geometrical and Topological Study of the Kosterlitz\u2013Thouless Phase Transition in the XY Model in Two Dimensions","volume":"2021","author":"Gori","year":"2021","journal-title":"J. Stat. Mech. Theory Exp."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"241","DOI":"10.1007\/BF01174334","article-title":"Inequalities of Willmore Type for Submanifolds","volume":"193","author":"Pinkall","year":"1986","journal-title":"Math. Z."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"385","DOI":"10.1216\/rmjm\/1030539623","article-title":"Fluctuation of Sectional Curvature for Closed Hypersurfaces","volume":"32","author":"Overholt","year":"2002","journal-title":"Rocky Mt. J. Math."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"285107","DOI":"10.1088\/0953-8984\/21\/28\/285107","article-title":"Dynamics and Energy Landscape in a Tetrahedral Network Glass-Former: Direct Comparison with Models of Fragile Liquids","volume":"21","author":"Coslovich","year":"2009","journal-title":"J. Phys. Condens. Matter"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"045102","DOI":"10.1103\/PhysRevE.63.045102","article-title":"Fast Monte Carlo Algorithm for Supercooled Soft Spheres","volume":"63","author":"Grigera","year":"2001","journal-title":"Phys. Rev. E"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"376","DOI":"10.1016\/j.physa.2018.10.001","article-title":"On the Origin of Phase Transitions in the Absence of Symmetry-Breaking","volume":"516","author":"Pettini","year":"2019","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"011127","DOI":"10.1103\/PhysRevE.84.011127","article-title":"Microcanonical Entropy Inflection Points: Key to Systematic Understanding of Transitions in Finite Systems","volume":"84","author":"Schnabel","year":"2011","journal-title":"Phys. Rev. E"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"180601","DOI":"10.1103\/PhysRevLett.120.180601","article-title":"Classification of Phase Transitions by Microcanonical Inflection-Point Analysis","volume":"120","author":"Qi","year":"2018","journal-title":"Phys. Rev. Lett."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"012013","DOI":"10.1088\/1742-6596\/487\/1\/012013","article-title":"Novel Concepts for the Systematic Statistical Analysis of Phase Transitions in Finite Systems","volume":"487","author":"Bachmann","year":"2014","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"4963","DOI":"10.1103\/PhysRevB.38.4963","article-title":"Critical Dynamics of Metallic Spin Glasses","volume":"38","year":"1988","journal-title":"Phys. Rev. B"},{"key":"ref_36","unstructured":"Chakraborty, T. (2024). Spin Glass Experiments. Encyclopedia of Condensed Matter Physics, Academic Press. [2nd ed.]."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1051\/jphys:0198600470107100","article-title":"Determination of the Critical Exponents in the Ag Mn Spin Glass","volume":"47","author":"Bouchiat","year":"1986","journal-title":"J. Phys."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Malthe-S\u00f8renssen, A. (2024). Finite Size Scaling. Percolation Theory Using Python, Springer International Publishing.","DOI":"10.1007\/978-3-031-59900-2"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"973","DOI":"10.1088\/0305-4470\/38\/5\/001","article-title":"Finite-Size Behaviour of the Microcanonical Specific Heat","volume":"38","author":"Behringer","year":"2005","journal-title":"J. Phys. Math. Gen."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"784","DOI":"10.1103\/PhysRevB.28.784","article-title":"Bond-Orientational Order in Liquids and Glasses","volume":"28","author":"Steinhardt","year":"1983","journal-title":"Phys. Rev. B"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"2256","DOI":"10.1063\/1.1532344","article-title":"Quantification of Order in the Lennard-Jones System","volume":"118","author":"Errington","year":"2003","journal-title":"J. Chem. Phys."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"154505","DOI":"10.1063\/1.3106759","article-title":"Mixing Effects in Glass-Forming Lennard-Jones Mixtures","volume":"130","author":"Valdes","year":"2009","journal-title":"J. Chem. Phys."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"993","DOI":"10.1103\/PhysRevE.62.993","article-title":"Towards a Quantification of Disorder in Materials: Distinguishing Equilibrium and Glassy Sphere Packings","volume":"62","author":"Truskett","year":"2000","journal-title":"Phys. Rev. E"},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Newman, M.E.J., and Barkema, G.T. (1999). Monte Carlo Methods in Statistical Physics, Oxford University Press.","DOI":"10.1093\/oso\/9780198517962.001.0001"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"4061","DOI":"10.1103\/PhysRevA.44.4061","article-title":"Microcanonical Ensemble Monte Carlo Method","volume":"44","author":"Ray","year":"1991","journal-title":"Phys. Rev. A"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"8816","DOI":"10.1063\/1.477552","article-title":"Microcanonical Monte Carlo Simulation of Thermodynamic Properties","volume":"109","author":"Lustig","year":"1998","journal-title":"J. Chem. Phys."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"3606","DOI":"10.1103\/PhysRevE.60.3606","article-title":"Domain-Wall Free Energy of Spin-Glass Models: Numerical Method and Boundary Conditions","volume":"60","author":"Hukushima","year":"1999","journal-title":"Phys. Rev. E"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"043311","DOI":"10.1103\/PhysRevE.100.043311","article-title":"Effects of Setting the Temperatures in the Parallel Tempering Monte Carlo Algorithm","volume":"100","author":"Rozada","year":"2019","journal-title":"Phys. Rev. E"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"5147","DOI":"10.1063\/1.445384","article-title":"Shear Viscosities Away from the Melting Line: A Comparison of Equilibrium and Nonequilibrium Molecular Dynamics","volume":"78","author":"Holian","year":"1983","journal-title":"J. Chem. Phys."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/3\/258\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T16:44:56Z","timestamp":1760028296000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/3\/258"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,2,28]]},"references-count":49,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2025,3]]}},"alternative-id":["e27030258"],"URL":"https:\/\/doi.org\/10.3390\/e27030258","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2025,2,28]]}}}