{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T00:53:58Z","timestamp":1776732838120,"version":"3.51.2"},"reference-count":45,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2020,3,26]],"date-time":"2020-03-26T00:00:00Z","timestamp":1585180800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In the present work, we discuss how the functional form of thermodynamic observables can be deduced from the geometric properties of subsets of phase space. The geometric quantities taken into account are mainly extrinsic curvatures of the energy level sets of the Hamiltonian of a system under investigation. In particular, it turns out that peculiar behaviours of thermodynamic observables at a phase transition point are rooted in more fundamental changes of the geometry of the energy level sets in phase space. More specifically, we discuss how microcanonical and geometrical descriptions of phase-transitions are shaped in the special case of     \u03d5 4     models with either nearest-neighbours and mean-field interactions.<\/jats:p>","DOI":"10.3390\/e22040380","type":"journal-article","created":{"date-parts":[[2020,3,27]],"date-time":"2020-03-27T09:04:38Z","timestamp":1585299878000},"page":"380","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["Geometrical Aspects in the Analysis of Microcanonical Phase-Transitions"],"prefix":"10.3390","volume":"22","author":[{"given":"Ghofrane","family":"Bel-Hadj-Aissa","sequence":"first","affiliation":[{"name":"Dipartimento di Scienze fisiche, della Terra e dell\u2019ambiente (DSFTA), University of Siena, Via Roma 56, 53100 Siena, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Matteo","family":"Gori","sequence":"additional","affiliation":[{"name":"Quantum Biology Lab, Howard University, 2400 6th St NW, Washington, DC 20059, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Vittorio","family":"Penna","sequence":"additional","affiliation":[{"name":"Dipartimento di Fisica, Politecnico di Torino, Corso Duca degli Abruzzi 24, I-10129 Torino, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Giulio","family":"Pettini","sequence":"additional","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"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7588-921X","authenticated-orcid":false,"given":"Roberto","family":"Franzosi","sequence":"additional","affiliation":[{"name":"QSTAR &amp; CNR\u2014Istituto Nazionale di Ottica, Largo Enrico Fermi 2, I-50125 Firenze, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,3,26]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"404","DOI":"10.1103\/PhysRev.87.404","article-title":"Statistical Theory of Equations of State and Phase Transitions. I. Theory of Condensation","volume":"87","author":"Yang","year":"1952","journal-title":"Phys. Rev."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Pettini, M. (2007). Geometry and Topology in Hamiltonian Dynamics and Statistical Mechanics, Springer.","DOI":"10.1007\/978-0-387-49957-4"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"4361","DOI":"10.1103\/PhysRevLett.79.4361","article-title":"Geometry of Dynamics, Lyapunov Exponents, and Phase Transitions","volume":"79","author":"Caiani","year":"1997","journal-title":"Phys. Rev. Lett."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"046205","DOI":"10.1103\/PhysRevE.78.046205","article-title":"Riemannian geometry of Hamiltonian chaos: Hints for a general theory","volume":"78","author":"Ciraolo","year":"2008","journal-title":"Phys. Rev. E"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"3886","DOI":"10.1103\/PhysRevE.57.3886","article-title":"Geometry of dynamics and phase transitions in classical lattice \u03c64 theories","volume":"57","author":"Caiani","year":"1998","journal-title":"Phys. Rev. E"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1016\/S0370-1573(00)00069-7","article-title":"Geometric approach to Hamiltonian dynamics and statistical mechanics","volume":"337","author":"Casetti","year":"2000","journal-title":"Phys. Rep."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"060601","DOI":"10.1103\/PhysRevLett.92.060601","article-title":"Theorem on the Origin of Phase Transitions","volume":"92","author":"Franzosi","year":"2004","journal-title":"Phys. Rev. Lett."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1016\/j.nuclphysb.2007.04.025","article-title":"Topology and phase transitions I. Preliminary results","volume":"782","author":"Franzosi","year":"2007","journal-title":"Nucl. Phys. B"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1016\/j.nuclphysb.2007.04.035","article-title":"Topology and phase transitions II. Theorem on a necessary relation","volume":"782","author":"Franzosi","year":"2007","journal-title":"Nucl. Phys. B"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"011303","DOI":"10.1103\/PhysRevE.85.011303","article-title":"Computational topology for configuration spaces of hard disks","volume":"85","author":"Carlsson","year":"2012","journal-title":"Phys. Rev. E"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"2577","DOI":"10.1093\/imrn\/rnt012","article-title":"Min-Type Morse Theory for Configuration Spaces of Hard Spheres","volume":"2014","author":"Baryshnikov","year":"2014","journal-title":"Int. Math. Res. Not."},{"key":"ref_12","first-page":"2021","article-title":"Quantum phase transitions without thermodynamic limits","volume":"463","author":"Brody","year":"2007","journal-title":"Proc. Roy. Soc. A (Lond.)"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"414","DOI":"10.1016\/j.aop.2016.10.017","article-title":"On the dispute between Boltzmann and Gibbs entropy","volume":"375","author":"Buonsante","year":"2016","journal-title":"Ann. Phys."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"052135","DOI":"10.1103\/PhysRevE.95.052135","article-title":"Phase transitions at high energy vindicate negative microcanonical temperature","volume":"95","author":"Buonsante","year":"2017","journal-title":"Phys. Rev. E"},{"key":"ref_15","first-page":"31","article-title":"Quantum phase transitions from topology in momentum space","volume":"Volume 463","author":"Volovik","year":"2007","journal-title":"Quantum Analogues: From Phase Transitions to Black Holes and Cosmology"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"055502","DOI":"10.1103\/PhysRevLett.87.055502","article-title":"Topological Description of the Aging Dynamics in Simple Glasses","volume":"87","author":"Angelani","year":"2001","journal-title":"Phys. Rev. Lett."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1038\/35065704","article-title":"Supercooled liquids and the glass transition","volume":"410","author":"Debenedetti","year":"2001","journal-title":"Nature"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"145702","DOI":"10.1103\/PhysRevLett.95.145702","article-title":"Topology, Phase Transitions, and the Spherical Model","volume":"95","author":"Stariolo","year":"2005","journal-title":"Phys. Rev. Lett."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"036125","DOI":"10.1103\/PhysRevE.70.036125","article-title":"Saddle index properties, singular topology, and its relation to thermodynamic singularities for a \u03d54 mean-field model","volume":"70","author":"Garanin","year":"2004","journal-title":"Phys. Rev. E"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"013202","DOI":"10.1088\/1742-5468\/2017\/1\/013202","article-title":"Topological approach to microcanonical thermodynamics and phase transition of interacting classical spins","volume":"2017","author":"Santos","year":"2017","journal-title":"J. Stat. Mech. Theory Exp."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"158102","DOI":"10.1103\/PhysRevLett.92.158102","article-title":"Topological Origin of the Phase Transition in a Model of DNA Denaturation","volume":"92","author":"Grinza","year":"2004","journal-title":"Phys. Rev. Lett."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1495","DOI":"10.1063\/1.473299","article-title":"The topology of multidimensional potential energy surfaces: Theory and application to peptide structure and kinetics","volume":"106","author":"Becker","year":"1997","journal-title":"J. Chem. Phys."},{"key":"ref_23","unstructured":"Huang, K. (1987). Statistical Mechanics, John Wiley & Sons. [2nd ed.]."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1038\/nphys2815","article-title":"Consistent thermostatistics forbids negative absolute temperatures","volume":"10","author":"Dunkel","year":"2013","journal-title":"Nat. Phys."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"302","DOI":"10.1016\/j.physa.2017.12.059","article-title":"Microcanonical entropy for classical systems","volume":"494","author":"Franzosi","year":"2018","journal-title":"Phys. A"},{"key":"ref_26","unstructured":"Khinchin, A. (1949). Mathematical Foundations of Statistical Mechanics, Dover Publications."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"772","DOI":"10.1103\/PhysRevLett.78.772","article-title":"Dynamical Approach to Temperature","volume":"78","author":"Rugh","year":"1997","journal-title":"Phys. Rev. Lett."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"824","DOI":"10.1007\/s10955-011-0200-4","article-title":"Microcanonical Entropy and Dynamical Measure of Temperature for Systems with Two First Integrals","volume":"143","author":"Franzosi","year":"2011","journal-title":"J. Stat. Phys."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"050101(R)","DOI":"10.1103\/PhysRevE.85.050101","article-title":"Geometric microcanonical thermodynamics for systems with first integrals","volume":"85","author":"Franzosi","year":"2012","journal-title":"Phys. Rev. E"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"R89","DOI":"10.1088\/0951-7715\/24\/12\/R01","article-title":"Discrete breathers in Bose\u2013Einstein condensates","volume":"24","author":"Franzosi","year":"2011","journal-title":"Nonlinearity"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"023032","DOI":"10.1088\/1367-2630\/15\/2\/023032","article-title":"Discrete breathers and negative-temperature states","volume":"15","author":"Iubini","year":"2013","journal-title":"New J. Phys."},{"key":"ref_32","unstructured":"Federer, H. (1969). Geometric Measure Theory, Springer. Die Grundlehren der mathematischen Wissenschaften, Band 153."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"258","DOI":"10.1007\/BF00945002","article-title":"On the convexity of geometric functional of level for solutions of certain elliptic partial differential equations","volume":"40","author":"Laurence","year":"1989","journal-title":"Z. Angew. Math. Phys."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"083204","DOI":"10.1088\/1742-5468\/ab3116","article-title":"A microcanonical entropy correcting finite-size effects in small systems","volume":"2019","author":"Franzosi","year":"2019","journal-title":"J. Stat. Mech. Theory Exp."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Gross, D. (2001). Microcanonical Thermodynamics: Phase Transitions in \"small\" Systems, World Scientific. World Scientific lecture notes in physics.","DOI":"10.1142\/4340"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"011120","DOI":"10.1103\/PhysRevE.74.011120","article-title":"Nonanalytic microscopic phase transitions and temperature oscillations in the microcanonical ensemble: An exactly solvable one-dimensional model for evaporation","volume":"74","author":"Hilbert","year":"2006","journal-title":"Phys. Rev. E"},{"key":"ref_37","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_38","doi-asserted-by":"crossref","first-page":"052148","DOI":"10.1103\/PhysRevE.93.052148","article-title":"Microcanonical analysis of a finite-size nonequilibrium system","volume":"93","author":"Lee","year":"2016","journal-title":"Phys. Rev. E"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"021301","DOI":"10.1103\/PhysRevE.94.021301","article-title":"First-order phase transitions in the real microcanonical ensemble","volume":"94","author":"Schierz","year":"2016","journal-title":"Phys. Rev. E"},{"key":"ref_40","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_41","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"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1088\/0031-8949\/51\/1\/005","article-title":"Efficient symplectic algorithms for numerical simulations of Hamiltonian flows","volume":"51","author":"Casetti","year":"1995","journal-title":"Phys. Scr."},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Bachmann, M. (2014). Thermodynamics and Statistical Mechanics of Macromolecular Systems, Cambridge University Press.","DOI":"10.1017\/CBO9781139028516"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"3907","DOI":"10.1103\/PhysRevD.43.3907","article-title":"Celestial mechanics, conformal structures, and gravitational waves","volume":"43","author":"Duval","year":"1991","journal-title":"Phys. Rev. D"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"591","DOI":"10.2307\/1968307","article-title":"Dynamical Trajectories and Geodesics","volume":"30","author":"Eisenhart","year":"1928","journal-title":"Ann. Math."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/4\/380\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:12:01Z","timestamp":1760173921000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/4\/380"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,3,26]]},"references-count":45,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2020,4]]}},"alternative-id":["e22040380"],"URL":"https:\/\/doi.org\/10.3390\/e22040380","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,3,26]]}}}