{"id":"https://openalex.org/W3208801559","doi":"https://doi.org/10.3390/e23111414","title":"Topology and Phase Transitions: A First Analytical Step towards the Definition of Sufficient Conditions","display_name":"Topology and Phase Transitions: A First Analytical Step towards the Definition of Sufficient Conditions","publication_year":2021,"publication_date":"2021-10-27","ids":{"openalex":"https://openalex.org/W3208801559","doi":"https://doi.org/10.3390/e23111414","mag":"3208801559","pmid":"https://pubmed.ncbi.nlm.nih.gov/34828112"},"language":"en","primary_location":{"id":"doi:10.3390/e23111414","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e23111414","pdf_url":"https://www.mdpi.com/1099-4300/23/11/1414/pdf?version=1635818936","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":"article","indexed_in":["crossref","datacite","doaj","pubmed"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/1099-4300/23/11/1414/pdf?version=1635818936","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5044878437","display_name":"Loris Di Cairano","orcid":"https://orcid.org/0000-0001-5859-4188"},"institutions":[{"id":"https://openalex.org/I171892758","display_name":"Forschungszentrum J\u00fclich","ror":"https://ror.org/02nv7yv05","country_code":"DE","type":"facility","lineage":["https://openalex.org/I1305996414","https://openalex.org/I171892758"]},{"id":"https://openalex.org/I887968799","display_name":"RWTH Aachen University","ror":"https://ror.org/04xfq0f34","country_code":"DE","type":"education","lineage":["https://openalex.org/I887968799"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Loris Di Cairano","raw_affiliation_strings":["Department of Physics, Faculty of Mathematics, Computer Science and Natural Sciences, Aachen University, 52062 Aachen, Germany","Institute of Neuroscience and Medicine INM-9, and Institute for Advanced Simulation IAS-5, Forschungszentrum J\u00fclich, 52428 J\u00fclich, Germany"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Physics, Faculty of Mathematics, Computer Science and Natural Sciences, Aachen University, 52062 Aachen, Germany","institution_ids":["https://openalex.org/I887968799"]},{"raw_affiliation_string":"Institute of Neuroscience and Medicine INM-9, and Institute for Advanced Simulation IAS-5, Forschungszentrum J\u00fclich, 52428 J\u00fclich, Germany","institution_ids":["https://openalex.org/I171892758"]}]},{"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":["Physics and Materials Science Research Unit, University of Luxembourg, L-1511 Luxembourg, Luxembourg"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Physics and Materials Science Research Unit, 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/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"],"is_corresponding":true,"raw_author_name":"Marco Pettini","raw_affiliation_strings":["Aix-Marseille Univ, CNRS, Universit\u00e9 de Toulon, 13288 Marseille, France","CNRS Centre de Physique Th\u00e9orique UMR7332, 13288 Marseille, France"],"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":"CNRS Centre de Physique Th\u00e9orique UMR7332, 13288 Marseille, France","institution_ids":["https://openalex.org/I4210160215","https://openalex.org/I1294671590"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5044783610"],"corresponding_institution_ids":["https://openalex.org/I1294671590","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":0.6558,"has_fulltext":true,"cited_by_count":8,"citation_normalized_percentile":{"value":0.68986341,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":99},"biblio":{"volume":"23","issue":"11","first_page":"1414","last_page":"1414"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11520","display_name":"Advanced Thermodynamics and Statistical Mechanics","score":0.9954000115394592,"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/T11520","display_name":"Advanced Thermodynamics and Statistical Mechanics","score":0.9954000115394592,"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":0.9909999966621399,"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/T10044","display_name":"Protein Structure and Dynamics","score":0.9901000261306763,"subfield":{"id":"https://openalex.org/subfields/1312","display_name":"Molecular Biology"},"field":{"id":"https://openalex.org/fields/13","display_name":"Biochemistry, Genetics and Molecular Biology"},"domain":{"id":"https://openalex.org/domains/1","display_name":"Life Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/differentiable-function","display_name":"Differentiable function","score":0.7029672861099243},{"id":"https://openalex.org/keywords/topology","display_name":"Topology (electrical circuits)","score":0.6924945712089539},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6260852813720703},{"id":"https://openalex.org/keywords/phase-space","display_name":"Phase space","score":0.5614776015281677},{"id":"https://openalex.org/keywords/configuration-space","display_name":"Configuration space","score":0.5578333139419556},{"id":"https://openalex.org/keywords/space","display_name":"Space (punctuation)","score":0.48704102635383606},{"id":"https://openalex.org/keywords/entropy","display_name":"Entropy (arrow of time)","score":0.481065034866333},{"id":"https://openalex.org/keywords/phase-transition","display_name":"Phase transition","score":0.4794929623603821},{"id":"https://openalex.org/keywords/topological-entropy","display_name":"Topological entropy","score":0.46495991945266724},{"id":"https://openalex.org/keywords/function-space","display_name":"Function space","score":0.4483998417854309},{"id":"https://openalex.org/keywords/class","display_name":"Class (philosophy)","score":0.4474669098854065},{"id":"https://openalex.org/keywords/differential-topology","display_name":"Differential topology","score":0.42687588930130005},{"id":"https://openalex.org/keywords/physical-space","display_name":"Physical space","score":0.4120686948299408},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.3191356956958771},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.26871126890182495},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.1815330982208252},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.10588619112968445},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.07455259561538696},{"id":"https://openalex.org/keywords/curvature","display_name":"Curvature","score":0.0634290874004364}],"concepts":[{"id":"https://openalex.org/C202615002","wikidata":"https://www.wikidata.org/wiki/Q783507","display_name":"Differentiable function","level":2,"score":0.7029672861099243},{"id":"https://openalex.org/C184720557","wikidata":"https://www.wikidata.org/wiki/Q7825049","display_name":"Topology (electrical circuits)","level":2,"score":0.6924945712089539},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6260852813720703},{"id":"https://openalex.org/C151342819","wikidata":"https://www.wikidata.org/wiki/Q62542","display_name":"Phase space","level":2,"score":0.5614776015281677},{"id":"https://openalex.org/C90738871","wikidata":"https://www.wikidata.org/wiki/Q41642869","display_name":"Configuration space","level":2,"score":0.5578333139419556},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.48704102635383606},{"id":"https://openalex.org/C106301342","wikidata":"https://www.wikidata.org/wiki/Q4117933","display_name":"Entropy (arrow of time)","level":2,"score":0.481065034866333},{"id":"https://openalex.org/C149288129","wikidata":"https://www.wikidata.org/wiki/Q185357","display_name":"Phase transition","level":2,"score":0.4794929623603821},{"id":"https://openalex.org/C2780350623","wikidata":"https://www.wikidata.org/wiki/Q3055355","display_name":"Topological entropy","level":2,"score":0.46495991945266724},{"id":"https://openalex.org/C142730499","wikidata":"https://www.wikidata.org/wiki/Q934367","display_name":"Function space","level":2,"score":0.4483998417854309},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.4474669098854065},{"id":"https://openalex.org/C516171914","wikidata":"https://www.wikidata.org/wiki/Q1224402","display_name":"Differential topology","level":5,"score":0.42687588930130005},{"id":"https://openalex.org/C2982717780","wikidata":"https://www.wikidata.org/wiki/Q107","display_name":"Physical space","level":2,"score":0.4120686948299408},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3191356956958771},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.26871126890182495},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.1815330982208252},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.10588619112968445},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.07455259561538696},{"id":"https://openalex.org/C195065555","wikidata":"https://www.wikidata.org/wiki/Q214881","display_name":"Curvature","level":2,"score":0.0634290874004364},{"id":"https://openalex.org/C111919701","wikidata":"https://www.wikidata.org/wiki/Q9135","display_name":"Operating system","level":1,"score":0.0},{"id":"https://openalex.org/C58640448","wikidata":"https://www.wikidata.org/wiki/Q42515","display_name":"Cartography","level":1,"score":0.0},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.0},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.0},{"id":"https://openalex.org/C12520029","wikidata":"https://www.wikidata.org/wiki/Q1147161","display_name":"Scalar curvature","level":3,"score":0.0},{"id":"https://openalex.org/C205649164","wikidata":"https://www.wikidata.org/wiki/Q1071","display_name":"Geography","level":0,"score":0.0},{"id":"https://openalex.org/C203934109","wikidata":"https://www.wikidata.org/wiki/Q7322952","display_name":"Ricci-flat manifold","level":4,"score":0.0}],"mesh":[],"locations_count":8,"locations":[{"id":"doi:10.3390/e23111414","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e23111414","pdf_url":"https://www.mdpi.com/1099-4300/23/11/1414/pdf?version=1635818936","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:34828112","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/34828112","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:HAL:hal-03540931v1","is_oa":true,"landing_page_url":"https://hal.science/hal-03540931","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Entropy, 2021, 23 (11), pp.1414. &#x27E8;10.3390/e23111414&#x27E9;","raw_type":"Journal articles"},{"id":"pmh:oai:doaj.org/article:06ad8109ccea43a0850aee98ab9eb293","is_oa":true,"landing_page_url":"https://doaj.org/article/06ad8109ccea43a0850aee98ab9eb293","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 23, Iss 11, p 1414 (2021)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/1099-4300/23/11/1414/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/e23111414","pdf_url":null,"source":{"id":"https://openalex.org/S4306400947","display_name":"MDPI (MDPI AG)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210097602","host_organization_name":"Multidisciplinary Digital Publishing Institute (Switzerland)","host_organization_lineage":["https://openalex.org/I4210097602"],"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; Volume 23; Issue 11; Pages: 1414","raw_type":"Text"},{"id":"pmh:oai:orbilu.uni.lu:10993/50206","is_oa":true,"landing_page_url":"https://orbilu.uni.lu/handle/10993/50206","pdf_url":null,"source":{"id":"https://openalex.org/S4306401815","display_name":"Open Repository and Bibliography (University of Luxembourg)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I186903577","host_organization_name":"University of Luxembourg","host_organization_lineage":["https://openalex.org/I186903577"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Entropy, 23 (11) (2021-10-27)","raw_type":"peer reviewed"},{"id":"pmh:oai:pubmedcentral.nih.gov:8621195","is_oa":true,"landing_page_url":"https://www.ncbi.nlm.nih.gov/pmc/articles/8621195","pdf_url":null,"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":"doi:10.18154/rwth-2022-00256","is_oa":true,"landing_page_url":"https://doi.org/10.18154/rwth-2022-00256","pdf_url":null,"source":{"id":"https://openalex.org/S4306401033","display_name":"RWTH Publications (RWTH Aachen)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I887968799","host_organization_name":"RWTH Aachen University","host_organization_lineage":["https://openalex.org/I887968799"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article-journal"}],"best_oa_location":{"id":"doi:10.3390/e23111414","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e23111414","pdf_url":"https://www.mdpi.com/1099-4300/23/11/1414/pdf?version=1635818936","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":[{"display_name":"Affordable and clean energy","id":"https://metadata.un.org/sdg/7","score":0.8700000047683716}],"awards":[],"funders":[{"id":"https://openalex.org/F4320309911","display_name":"Aix-Marseille Universit\u00e9","ror":"https://ror.org/035xkbk20"},{"id":"https://openalex.org/F4320309912","display_name":"Fondation Aix-Marseille Universite","ror":"https://ror.org/035xkbk20"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W3208801559.pdf","grobid_xml":"https://content.openalex.org/works/W3208801559.grobid-xml"},"referenced_works_count":37,"referenced_works":["https://openalex.org/W1507653385","https://openalex.org/W1524852997","https://openalex.org/W1634908474","https://openalex.org/W1964622138","https://openalex.org/W1965603942","https://openalex.org/W1966867233","https://openalex.org/W1983566626","https://openalex.org/W2009768419","https://openalex.org/W2017865506","https://openalex.org/W2023684743","https://openalex.org/W2038026589","https://openalex.org/W2039470150","https://openalex.org/W2068565660","https://openalex.org/W2070690215","https://openalex.org/W2074391965","https://openalex.org/W2121536219","https://openalex.org/W2227075066","https://openalex.org/W2325291401","https://openalex.org/W2753312321","https://openalex.org/W2772729974","https://openalex.org/W2798740381","https://openalex.org/W2802689177","https://openalex.org/W2951420393","https://openalex.org/W2963174906","https://openalex.org/W2964269906","https://openalex.org/W2968995995","https://openalex.org/W2995671148","https://openalex.org/W2998283595","https://openalex.org/W3006557790","https://openalex.org/W3098717111","https://openalex.org/W3098717765","https://openalex.org/W3103960812","https://openalex.org/W3131150476","https://openalex.org/W3149216993","https://openalex.org/W4239702435","https://openalex.org/W4243132536","https://openalex.org/W4250933756"],"related_works":["https://openalex.org/W4246475047","https://openalex.org/W2622208885","https://openalex.org/W2037328585","https://openalex.org/W1983457500","https://openalex.org/W2129584172","https://openalex.org/W2297004757","https://openalex.org/W2144524562","https://openalex.org/W2349517432","https://openalex.org/W1987144560","https://openalex.org/W2809106743"],"abstract_inverted_index":{"Different":[0],"arguments":[1],"led":[2],"to":[3,13],"supposing":[4],"that":[5],"the":[6,65,69,116,125,129,146],"deep":[7],"origin":[8],"of":[9,20,24,27,49,60,64,107,120,124,128,143,145],"phase":[10,52,66,130],"transitions":[11,53],"has":[12],"be":[14,136],"identified":[15],"with":[16,40],"suitable":[17],"topological":[18,58],"changes":[19,59],"potential":[21],"related":[22],"submanifolds":[23,63,127],"configuration":[25],"space":[26],"a":[28,46,74,81,93,105,121],"physical":[29,50],"system.":[30],"An":[31],"important":[32,83],"step":[33,84],"forward":[34,85],"for":[35,45,140],"this":[36,79,89,111],"approach":[37],"was":[38,86,96],"achieved":[39],"two":[41],"theorems":[42],"stating":[43],"that,":[44],"wide":[47,75],"class":[48],"systems,":[51],"should":[54],"necessarily":[55],"stem":[56],"from":[57],"energy":[61,118],"level":[62],"space.":[67,131],"However,":[68],"sufficiency":[70],"conditions":[71,134],"are":[72],"still":[73],"open":[76],"question.":[77],"In":[78],"study,":[80],"first":[82],"performed":[87],"in":[88,91,137],"direction;":[90],"fact,":[92],"differential":[94],"equation":[95],"worked":[97],"out":[98],"which":[99],"describes":[100],"how":[101],"entropy":[102],"varies":[103],"as":[104],"function":[106],"total":[108,117],"energy,":[109],"and":[110],"variation":[112],"is":[113],"driven":[114],"by":[115],"dependence":[119],"topology-related":[122],"quantity":[123],"relevant":[126],"Hence,":[132],"general":[133],"can":[135],"principle":[138],"defined":[139],"topology-driven":[141],"loss":[142],"differentiability":[144],"entropy.":[147]},"counts_by_year":[{"year":2026,"cited_by_count":3},{"year":2024,"cited_by_count":2},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":2}],"updated_date":"2026-05-21T09:19:25.381259","created_date":"2021-11-08T00:00:00"}
