{"id":"https://openalex.org/W4403190459","doi":"https://doi.org/10.3390/e26100840","title":"From Geometry of Hamiltonian Dynamics to Topology of Phase Transitions: A Review","display_name":"From Geometry of Hamiltonian Dynamics to Topology of Phase Transitions: A Review","publication_year":2024,"publication_date":"2024-10-05","ids":{"openalex":"https://openalex.org/W4403190459","doi":"https://doi.org/10.3390/e26100840","pmid":"https://pubmed.ncbi.nlm.nih.gov/39451917"},"language":"en","primary_location":{"id":"doi:10.3390/e26100840","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e26100840","pdf_url":"https://www.mdpi.com/1099-4300/26/10/840/pdf?version=1728099998","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy","raw_type":"journal-article"},"type":"review","indexed_in":["crossref","doaj","pubmed"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/1099-4300/26/10/840/pdf?version=1728099998","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5016071235","display_name":"Giulio Pettini","orcid":"https://orcid.org/0000-0002-8045-4875"},"institutions":[{"id":"https://openalex.org/I4210133295","display_name":"Istituto Nazionale di Fisica Nucleare, Sezione di Firenze","ror":"https://ror.org/02vv5y108","country_code":"IT","type":"facility","lineage":["https://openalex.org/I160013858","https://openalex.org/I4210133295"]}],"countries":["IT"],"is_corresponding":false,"raw_author_name":"Giulio Pettini","raw_affiliation_strings":["Dipartimento di Fisica, Universit\u00e0 di Firenze, and I.N.F.N., Sezione di Firenze, via G. Sansone 1, I-50019 Sesto Fiorentino, Italy"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Dipartimento di Fisica, Universit\u00e0 di Firenze, and I.N.F.N., Sezione di Firenze, via G. Sansone 1, I-50019 Sesto Fiorentino, Italy","institution_ids":["https://openalex.org/I4210133295"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5068569453","display_name":"Matteo Gori","orcid":"https://orcid.org/0000-0002-0777-133X"},"institutions":[{"id":"https://openalex.org/I186903577","display_name":"University of Luxembourg","ror":"https://ror.org/036x5ad56","country_code":"LU","type":"education","lineage":["https://openalex.org/I186903577"]}],"countries":["LU"],"is_corresponding":false,"raw_author_name":"Matteo Gori","raw_affiliation_strings":["Department of Physics and Sciences of Materials, University of Luxembourg, L-1511 Luxembourg, Luxembourg"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Physics and Sciences of Materials, University of Luxembourg, L-1511 Luxembourg, Luxembourg","institution_ids":["https://openalex.org/I186903577"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5044783610","display_name":"Marco Pettini","orcid":"https://orcid.org/0000-0003-1561-4390"},"institutions":[{"id":"https://openalex.org/I1294671590","display_name":"Centre National de la Recherche Scientifique","ror":"https://ror.org/02feahw73","country_code":"FR","type":"government","lineage":["https://openalex.org/I1294671590"]},{"id":"https://openalex.org/I137853757","display_name":"Howard University","ror":"https://ror.org/05gt1vc06","country_code":"US","type":"education","lineage":["https://openalex.org/I137853757"]},{"id":"https://openalex.org/I143002897","display_name":"Universit\u00e9 de Toulon","ror":"https://ror.org/02m9kbe37","country_code":"FR","type":"education","lineage":["https://openalex.org/I143002897"]},{"id":"https://openalex.org/I4210160215","display_name":"Centre de Physique Th\u00e9orique","ror":"https://ror.org/052bbtn31","country_code":"FR","type":"facility","lineage":["https://openalex.org/I1294671590","https://openalex.org/I1294671590","https://openalex.org/I143002897","https://openalex.org/I21491767","https://openalex.org/I4210098836","https://openalex.org/I4210160215"]}],"countries":["FR","US"],"is_corresponding":true,"raw_author_name":"Marco Pettini","raw_affiliation_strings":["Aix-Marseille Univ, CNRS, Universit\u00e9 de Toulon, 13288 Marseille, France","Centre de Physique Th\u00e9orique, 13288 Marseille, France","Quantum Biology Lab, Howard University, Washington, DC 20059, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Aix-Marseille Univ, CNRS, Universit\u00e9 de Toulon, 13288 Marseille, France","institution_ids":["https://openalex.org/I143002897","https://openalex.org/I1294671590"]},{"raw_affiliation_string":"Centre de Physique Th\u00e9orique, 13288 Marseille, France","institution_ids":["https://openalex.org/I4210160215"]},{"raw_affiliation_string":"Quantum Biology Lab, Howard University, Washington, DC 20059, USA","institution_ids":["https://openalex.org/I137853757"]}]}],"institutions":[],"countries_distinct_count":4,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5044783610"],"corresponding_institution_ids":["https://openalex.org/I1294671590","https://openalex.org/I137853757","https://openalex.org/I143002897","https://openalex.org/I4210160215"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.15538643,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"26","issue":"10","first_page":"840","last_page":"840"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11261","display_name":"Quantum chaos and dynamical systems","score":0.9994999766349792,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11261","display_name":"Quantum chaos and dynamical systems","score":0.9994999766349792,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12261","display_name":"Statistical Mechanics and Entropy","score":0.996399998664856,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11520","display_name":"Advanced Thermodynamics and Statistical Mechanics","score":0.9925000071525574,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/hamiltonian-system","display_name":"Hamiltonian system","score":0.6787487864494324},{"id":"https://openalex.org/keywords/geodesic","display_name":"Geodesic","score":0.6735010743141174},{"id":"https://openalex.org/keywords/lyapunov-exponent","display_name":"Lyapunov exponent","score":0.6533684134483337},{"id":"https://openalex.org/keywords/phase-transition","display_name":"Phase transition","score":0.6281372308731079},{"id":"https://openalex.org/keywords/phase-space","display_name":"Phase space","score":0.5807833075523376},{"id":"https://openalex.org/keywords/hamiltonian","display_name":"Hamiltonian (control theory)","score":0.5651630163192749},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.4829111099243164},{"id":"https://openalex.org/keywords/dynamical-systems-theory","display_name":"Dynamical systems theory","score":0.4533320367336273},{"id":"https://openalex.org/keywords/topology","display_name":"Topology (electrical circuits)","score":0.4484531581401825},{"id":"https://openalex.org/keywords/riemannian-geometry","display_name":"Riemannian geometry","score":0.43537068367004395},{"id":"https://openalex.org/keywords/classical-mechanics","display_name":"Classical mechanics","score":0.4067455530166626},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.37000396847724915},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.2835318148136139},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.14549791812896729},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.1023721992969513}],"concepts":[{"id":"https://openalex.org/C121770821","wikidata":"https://www.wikidata.org/wiki/Q2072471","display_name":"Hamiltonian system","level":2,"score":0.6787487864494324},{"id":"https://openalex.org/C165818556","wikidata":"https://www.wikidata.org/wiki/Q213488","display_name":"Geodesic","level":2,"score":0.6735010743141174},{"id":"https://openalex.org/C191544260","wikidata":"https://www.wikidata.org/wiki/Q1238630","display_name":"Lyapunov exponent","level":3,"score":0.6533684134483337},{"id":"https://openalex.org/C149288129","wikidata":"https://www.wikidata.org/wiki/Q185357","display_name":"Phase transition","level":2,"score":0.6281372308731079},{"id":"https://openalex.org/C151342819","wikidata":"https://www.wikidata.org/wiki/Q62542","display_name":"Phase space","level":2,"score":0.5807833075523376},{"id":"https://openalex.org/C130787639","wikidata":"https://www.wikidata.org/wiki/Q5645293","display_name":"Hamiltonian (control theory)","level":2,"score":0.5651630163192749},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.4829111099243164},{"id":"https://openalex.org/C79379906","wikidata":"https://www.wikidata.org/wiki/Q3174497","display_name":"Dynamical systems theory","level":2,"score":0.4533320367336273},{"id":"https://openalex.org/C184720557","wikidata":"https://www.wikidata.org/wiki/Q7825049","display_name":"Topology (electrical circuits)","level":2,"score":0.4484531581401825},{"id":"https://openalex.org/C181104567","wikidata":"https://www.wikidata.org/wiki/Q761383","display_name":"Riemannian geometry","level":2,"score":0.43537068367004395},{"id":"https://openalex.org/C74650414","wikidata":"https://www.wikidata.org/wiki/Q11397","display_name":"Classical mechanics","level":1,"score":0.4067455530166626},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.37000396847724915},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.2835318148136139},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.14549791812896729},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.1023721992969513},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.0},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.0}],"mesh":[],"locations_count":6,"locations":[{"id":"doi:10.3390/e26100840","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e26100840","pdf_url":"https://www.mdpi.com/1099-4300/26/10/840/pdf?version=1728099998","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy","raw_type":"journal-article"},{"id":"pmid:39451917","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/39451917","pdf_url":null,"source":{"id":"https://openalex.org/S4306525036","display_name":"PubMed","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy (Basel, Switzerland)","raw_type":null},{"id":"pmh:oai:pubmedcentral.nih.gov:11507261","is_oa":true,"landing_page_url":"https://www.ncbi.nlm.nih.gov/pmc/articles/11507261","pdf_url":"https://pmc.ncbi.nlm.nih.gov/articles/PMC11507261/pdf/entropy-26-00840.pdf","source":{"id":"https://openalex.org/S2764455111","display_name":"PubMed Central","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Entropy (Basel)","raw_type":"Text"},{"id":"pmh:oai:HAL:hal-04786053v1","is_oa":false,"landing_page_url":"https://hal.science/hal-04786053","pdf_url":null,"source":{"id":"https://openalex.org/S4306402512","display_name":"HAL (Le Centre pour la Communication Scientifique Directe)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1294671590","host_organization_name":"Centre National de la Recherche Scientifique","host_organization_lineage":["https://openalex.org/I1294671590"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Entropy, 2024, 26 (10), pp.840. &#x27E8;10.3390/e26100840&#x27E9;","raw_type":"Journal articles"},{"id":"pmh:oai:doaj.org/article:a24a35ff5e7b4c02b6abdf02f0308224","is_oa":true,"landing_page_url":"https://doaj.org/article/a24a35ff5e7b4c02b6abdf02f0308224","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Entropy, Vol 26, Iss 10, p 840 (2024)","raw_type":"article"},{"id":"pmh:oai:flore.unifi.it:2158/1401324","is_oa":true,"landing_page_url":"https://www.mdpi.com/1099-4300/26/10/840","pdf_url":null,"source":{"id":"https://openalex.org/S4306402033","display_name":"Florence Research (University of Florence)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I45084792","host_organization_name":"University of Florence","host_organization_lineage":["https://openalex.org/I45084792"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"info:eu-repo/semantics/article"}],"best_oa_location":{"id":"doi:10.3390/e26100840","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e26100840","pdf_url":"https://www.mdpi.com/1099-4300/26/10/840/pdf?version=1728099998","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy","raw_type":"journal-article"},"sustainable_development_goals":[{"score":0.4300000071525574,"id":"https://metadata.un.org/sdg/7","display_name":"Affordable and clean energy"}],"awards":[{"id":"https://openalex.org/G7871603482","display_name":"Long-range electrodynamic INteractions between proteinS","funder_award_id":"964203","funder_id":"https://openalex.org/F4320320300","funder_display_name":"European Commission"}],"funders":[{"id":"https://openalex.org/F4320320300","display_name":"European Commission","ror":"https://ror.org/00k4n6c32"}],"has_content":{"grobid_xml":false,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4403190459.pdf"},"referenced_works_count":78,"referenced_works":["https://openalex.org/W634863351","https://openalex.org/W1524852997","https://openalex.org/W1636598878","https://openalex.org/W1733502595","https://openalex.org/W1963871440","https://openalex.org/W1965603942","https://openalex.org/W1966361272","https://openalex.org/W1966867233","https://openalex.org/W1968684655","https://openalex.org/W1971887382","https://openalex.org/W1980253839","https://openalex.org/W1981765609","https://openalex.org/W1989571057","https://openalex.org/W1994006347","https://openalex.org/W2001084352","https://openalex.org/W2008317504","https://openalex.org/W2009768419","https://openalex.org/W2017865506","https://openalex.org/W2023684743","https://openalex.org/W2026850275","https://openalex.org/W2028591115","https://openalex.org/W2036487145","https://openalex.org/W2037274336","https://openalex.org/W2038026589","https://openalex.org/W2039470150","https://openalex.org/W2042935608","https://openalex.org/W2047591624","https://openalex.org/W2053296163","https://openalex.org/W2053750969","https://openalex.org/W2065864578","https://openalex.org/W2068565660","https://openalex.org/W2072089840","https://openalex.org/W2081675404","https://openalex.org/W2083531734","https://openalex.org/W2087098251","https://openalex.org/W2090602755","https://openalex.org/W2104205738","https://openalex.org/W2104549471","https://openalex.org/W2114479908","https://openalex.org/W2121536219","https://openalex.org/W2121732682","https://openalex.org/W2124784909","https://openalex.org/W2125986857","https://openalex.org/W2129010625","https://openalex.org/W2129020329","https://openalex.org/W2149651960","https://openalex.org/W2157528880","https://openalex.org/W2172264954","https://openalex.org/W2227075066","https://openalex.org/W2295059836","https://openalex.org/W2312378499","https://openalex.org/W2325291401","https://openalex.org/W2469387292","https://openalex.org/W2753312321","https://openalex.org/W2798740381","https://openalex.org/W2802689177","https://openalex.org/W2901805157","https://openalex.org/W2951420393","https://openalex.org/W2964269906","https://openalex.org/W2978669281","https://openalex.org/W2995671148","https://openalex.org/W2998283595","https://openalex.org/W3022165290","https://openalex.org/W3098065780","https://openalex.org/W3100297669","https://openalex.org/W3101497996","https://openalex.org/W3105661112","https://openalex.org/W3162324712","https://openalex.org/W3208801559","https://openalex.org/W4205727720","https://openalex.org/W4214497182","https://openalex.org/W4225352806","https://openalex.org/W4226111999","https://openalex.org/W4235840289","https://openalex.org/W4301652612","https://openalex.org/W4308965807","https://openalex.org/W6647376483","https://openalex.org/W6671444144"],"related_works":["https://openalex.org/W2973136064","https://openalex.org/W3105747609","https://openalex.org/W2012691558","https://openalex.org/W2009385573","https://openalex.org/W1990019600","https://openalex.org/W2161938318","https://openalex.org/W1486814401","https://openalex.org/W4293874102","https://openalex.org/W2130044423","https://openalex.org/W2991226590"],"abstract_inverted_index":{"In":[0,67,206],"this":[1],"review":[2],"work,":[3],"we":[4],"outline":[5],"a":[6,76,85,268,283],"conceptual":[7],"path":[8],"that,":[9],"starting":[10],"from":[11],"the":[12,16,37,49,54,69,96,100,107,112,115,137,144,147,158,164,169,183,190,199,204,212,216,223,228,231,240,244,247,261],"numerical":[13,70],"investigation":[14,71,145],"of":[15,30,39,51,53,57,72,75,79,81,99,106,109,111,121,126,139,146,150,157,185,189,198,215,230,239,246,255,260,264],"transition":[17,88,117,171,225],"between":[18],"weak":[19],"chaos":[20,23],"and":[21,294],"strong":[22],"in":[24,48,124,161],"Hamiltonian":[25,58,73,122],"systems":[26,59,62],"with":[27,143,288],"many":[28],"degrees":[29,80],"freedom,":[31],"comes":[32],"to":[33,135,163,237,253,271,282,295],"highlight":[34],"how,":[35],"at":[36,114,168,221],"basis":[38],"equilibrium":[40],"phase":[41,55,65,87,116,151,170,224,265,273],"transitions,":[42],"there":[43],"must":[44],"be":[45,280],"major":[46],"changes":[47,156,167,181,188],"topology":[50],"submanifolds":[52],"space":[56,192],"that":[60,63,83,104,177,275],"describe":[61],"exhibit":[64],"transitions.":[66],"fact,":[68],"flows":[74,123,128],"large":[77],"number":[78],"freedom":[82],"undergo":[84],"thermodynamic":[86],"has":[89],"revealed":[90],"peculiar":[91,154,165,179],"dynamical":[92,148,166],"signatures":[93],"detected":[94],"through":[95],"energy":[97,219],"dependence":[98],"largest":[101],"Lyapunov":[102],"exponent,":[103],"is,":[105,287],"degree":[108],"chaoticity":[110],"dynamics":[113],"point.":[118,172],"The":[119,257],"geometrization":[120],"terms":[125],"geodesic":[127],"on":[129],"suitably":[130],"defined":[131],"Riemannian":[132],"manifolds,":[133],"used":[134],"explain":[136],"origin":[138,263],"deterministic":[140],"chaos,":[141],"combined":[142],"counterpart":[149],"transitions":[152,266,274],"unveils":[153],"geometrical":[155,180],"mechanical":[159],"manifolds":[160,200],"correspondence":[162],"Then,":[173],"it":[174],"turns":[175],"out":[176],"these":[178],"are":[182,234,250],"effect":[184],"deeper":[186],"topological":[187,258],"configuration":[191],"hypersurfaces":[193],"\u2211v=VN\u22121(v)":[194],"as":[195,197],"well":[196],"{Mv=VN\u22121((\u2212\u221e,v])}v\u2208R":[201],"bounded":[202],"by":[203,210],"\u2211v.":[205],"other":[207],"words,":[208],"denoting":[209],"vc":[211],"critical":[213],"value":[214],"average":[217],"potential":[218],"density":[220],"which":[222],"takes":[226],"place,":[227],"members":[229,245],"family":[232,241,248],"{\u2211v}v&lt;vc":[233],"not":[235,251,279],"diffeomorphic":[236,252],"those":[238,254],"{\u2211v}v&gt;vc;":[242],"additionally,":[243],"{Mv}v&gt;vc":[249],"{Mv}v&gt;vc.":[256],"theory":[259],"deep":[262],"allows":[267],"unifying":[269],"framework":[270],"tackle":[272],"may":[276,278],"or":[277,289],"due":[281],"symmetry-breaking":[284],"phenomenon":[285],"(that":[286],"without":[290],"an":[291],"order":[292],"parameter)":[293],"finite/small":[296],"N":[297],"systems.":[298]},"counts_by_year":[],"updated_date":"2026-05-21T09:19:25.381259","created_date":"2025-10-10T00:00:00"}
