{"id":"https://openalex.org/W2986942933","doi":"https://doi.org/10.3390/e21121155","title":"M\u00f6bius Transforms, Cycles and q-triplets in Statistical Mechanics","display_name":"M\u00f6bius Transforms, Cycles and q-triplets in Statistical Mechanics","publication_year":2019,"publication_date":"2019-11-26","ids":{"openalex":"https://openalex.org/W2986942933","doi":"https://doi.org/10.3390/e21121155","mag":"2986942933"},"language":"en","primary_location":{"id":"doi:10.3390/e21121155","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e21121155","pdf_url":"https://www.mdpi.com/1099-4300/21/12/1155/pdf?version=1577337052","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":["arxiv","crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/1099-4300/21/12/1155/pdf?version=1577337052","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5063248731","display_name":"Jean\u2010Pierre Gazeau","orcid":"https://orcid.org/0000-0001-7681-7672"},"institutions":[{"id":"https://openalex.org/I204730241","display_name":"Universit\u00e9 Paris Cit\u00e9","ror":"https://ror.org/05f82e368","country_code":"FR","type":"education","lineage":["https://openalex.org/I204730241"]},{"id":"https://openalex.org/I2802090066","display_name":"Laboratoire AstroParticule et Cosmologie","ror":"https://ror.org/03tnjrr49","country_code":"FR","type":"facility","lineage":["https://openalex.org/I1294671590","https://openalex.org/I1294671590","https://openalex.org/I204730241","https://openalex.org/I2738703131","https://openalex.org/I2746051580","https://openalex.org/I2799535048","https://openalex.org/I2800004676","https://openalex.org/I2802090066","https://openalex.org/I4210133362"]},{"id":"https://openalex.org/I4210091437","display_name":"Sorbonne Paris Cit\u00e9","ror":"https://ror.org/001z21q04","country_code":"FR","type":"other","lineage":["https://openalex.org/I4210091437"]},{"id":"https://openalex.org/I4210125245","display_name":"Centro Brasileiro de Pesquisas F\u00edsicas","ror":"https://ror.org/02wnmk332","country_code":"BR","type":"facility","lineage":["https://openalex.org/I4210125245","https://openalex.org/I4210151455"]}],"countries":["BR","FR"],"is_corresponding":false,"raw_author_name":"Jean Pierre Gazeau","raw_affiliation_strings":["Astroparticles and Cosmology (UMR 7164), Sorbonne Paris Cit\u00e9, Univ Paris Diderot, 75205 Paris, France","Centro Brasileiro de Pesquisas F\u00edsicas and National Institute of Science and Technology of Complex Systems, Rua Xavier Sigaud 150, Rio de Janeiro 22290-180, Brazil"],"raw_orcid":"https://orcid.org/0000-0001-7681-7672","affiliations":[{"raw_affiliation_string":"Astroparticles and Cosmology (UMR 7164), Sorbonne Paris Cit\u00e9, Univ Paris Diderot, 75205 Paris, France","institution_ids":["https://openalex.org/I204730241","https://openalex.org/I2802090066","https://openalex.org/I4210091437"]},{"raw_affiliation_string":"Centro Brasileiro de Pesquisas F\u00edsicas and National Institute of Science and Technology of Complex Systems, Rua Xavier Sigaud 150, Rio de Janeiro 22290-180, Brazil","institution_ids":["https://openalex.org/I4210125245"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5051451815","display_name":"Constantino Tsallis","orcid":"https://orcid.org/0000-0002-9387-9194"},"institutions":[{"id":"https://openalex.org/I1308548392","display_name":"Santa Fe Institute","ror":"https://ror.org/01arysc35","country_code":"US","type":"funder","lineage":["https://openalex.org/I1308548392"]},{"id":"https://openalex.org/I4210112164","display_name":"Complexity Science Hub","ror":"https://ror.org/023dz9m50","country_code":"AT","type":"nonprofit","lineage":["https://openalex.org/I4210112164"]},{"id":"https://openalex.org/I4210125245","display_name":"Centro Brasileiro de Pesquisas F\u00edsicas","ror":"https://ror.org/02wnmk332","country_code":"BR","type":"facility","lineage":["https://openalex.org/I4210125245","https://openalex.org/I4210151455"]}],"countries":["AT","BR","US"],"is_corresponding":true,"raw_author_name":"Constantino Tsallis","raw_affiliation_strings":["Centro Brasileiro de Pesquisas F\u00edsicas and National Institute of Science and Technology of Complex Systems, Rua Xavier Sigaud 150, Rio de Janeiro 22290-180, Brazil","Complexity Science Hub Vienna, Josefst\u00e4dter Strasse 39, 1080 Vienna, Austria","Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA"],"raw_orcid":"https://orcid.org/0000-0002-9387-9194","affiliations":[{"raw_affiliation_string":"Centro Brasileiro de Pesquisas F\u00edsicas and National Institute of Science and Technology of Complex Systems, Rua Xavier Sigaud 150, Rio de Janeiro 22290-180, Brazil","institution_ids":["https://openalex.org/I4210125245"]},{"raw_affiliation_string":"Complexity Science Hub Vienna, Josefst\u00e4dter Strasse 39, 1080 Vienna, Austria","institution_ids":["https://openalex.org/I4210112164"]},{"raw_affiliation_string":"Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA","institution_ids":["https://openalex.org/I1308548392"]}]}],"institutions":[],"countries_distinct_count":4,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5051451815"],"corresponding_institution_ids":["https://openalex.org/I1308548392","https://openalex.org/I4210112164","https://openalex.org/I4210125245"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":1.4415,"has_fulltext":true,"cited_by_count":15,"citation_normalized_percentile":{"value":0.82521276,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":94,"max":98},"biblio":{"volume":"21","issue":"12","first_page":"1155","last_page":"1155"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12261","display_name":"Statistical Mechanics and Entropy","score":0.9998000264167786,"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/T12261","display_name":"Statistical Mechanics and Entropy","score":0.9998000264167786,"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/T11871","display_name":"Advanced Statistical Methods and Models","score":0.9624000191688538,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10288","display_name":"Fractional Differential Equations Solutions","score":0.9573000073432922,"subfield":{"id":"https://openalex.org/subfields/2611","display_name":"Modeling and Simulation"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/multiplicative-function","display_name":"Multiplicative function","score":0.7372392416000366},{"id":"https://openalex.org/keywords/statistical-mechanics","display_name":"Statistical mechanics","score":0.6926777362823486},{"id":"https://openalex.org/keywords/observable","display_name":"Observable","score":0.6884363293647766},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6210063695907593},{"id":"https://openalex.org/keywords/boltzmann-constant","display_name":"Boltzmann constant","score":0.49427056312561035},{"id":"https://openalex.org/keywords/statistical-physics","display_name":"Statistical physics","score":0.3850848972797394},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.36909496784210205},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.33777564764022827},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.28187042474746704},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.22359421849250793},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.22209948301315308}],"concepts":[{"id":"https://openalex.org/C42747912","wikidata":"https://www.wikidata.org/wiki/Q1048447","display_name":"Multiplicative function","level":2,"score":0.7372392416000366},{"id":"https://openalex.org/C99874945","wikidata":"https://www.wikidata.org/wiki/Q188715","display_name":"Statistical mechanics","level":2,"score":0.6926777362823486},{"id":"https://openalex.org/C32848918","wikidata":"https://www.wikidata.org/wiki/Q845789","display_name":"Observable","level":2,"score":0.6884363293647766},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6210063695907593},{"id":"https://openalex.org/C35304006","wikidata":"https://www.wikidata.org/wiki/Q5962","display_name":"Boltzmann constant","level":2,"score":0.49427056312561035},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.3850848972797394},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.36909496784210205},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.33777564764022827},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.28187042474746704},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.22359421849250793},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.22209948301315308}],"mesh":[],"locations_count":5,"locations":[{"id":"doi:10.3390/e21121155","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e21121155","pdf_url":"https://www.mdpi.com/1099-4300/21/12/1155/pdf?version=1577337052","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":"pmh:oai:arXiv.org:1911.00594","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1911.00594","pdf_url":"https://arxiv.org/pdf/1911.00594","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"pmh:oai:HAL:hal-04301978v1","is_oa":false,"landing_page_url":"https://hal.science/hal-04301978","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, 2019, 21 (12), pp.1155. &#x27E8;10.3390/e21121155&#x27E9;","raw_type":"Journal articles"},{"id":"pmh:oai:mdpi.com:/1099-4300/21/12/1155/","is_oa":true,"landing_page_url":"http://dx.doi.org/10.3390/e21121155","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","raw_type":"Text"},{"id":"pmh:oai:pubmedcentral.nih.gov:7514500","is_oa":true,"landing_page_url":"https://www.ncbi.nlm.nih.gov/pmc/articles/7514500","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"}],"best_oa_location":{"id":"doi:10.3390/e21121155","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e21121155","pdf_url":"https://www.mdpi.com/1099-4300/21/12/1155/pdf?version=1577337052","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":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2986942933.pdf","grobid_xml":"https://content.openalex.org/works/W2986942933.grobid-xml"},"referenced_works_count":38,"referenced_works":["https://openalex.org/W1600090841","https://openalex.org/W1822373189","https://openalex.org/W1892782243","https://openalex.org/W1968110842","https://openalex.org/W1974819641","https://openalex.org/W1976719090","https://openalex.org/W1977485510","https://openalex.org/W1980097479","https://openalex.org/W1983874169","https://openalex.org/W1991755147","https://openalex.org/W1997414622","https://openalex.org/W2003153071","https://openalex.org/W2003703127","https://openalex.org/W2005619765","https://openalex.org/W2017963899","https://openalex.org/W2042342956","https://openalex.org/W2048721659","https://openalex.org/W2052156030","https://openalex.org/W2087965488","https://openalex.org/W2089852843","https://openalex.org/W2091585247","https://openalex.org/W2128633210","https://openalex.org/W2133969428","https://openalex.org/W2240510098","https://openalex.org/W2270028664","https://openalex.org/W2283358846","https://openalex.org/W2324785233","https://openalex.org/W2497099222","https://openalex.org/W2513141401","https://openalex.org/W2564025328","https://openalex.org/W2773633323","https://openalex.org/W2802491245","https://openalex.org/W2963169885","https://openalex.org/W3100556136","https://openalex.org/W3101281679","https://openalex.org/W3105508676","https://openalex.org/W4206509365","https://openalex.org/W4249046827"],"related_works":["https://openalex.org/W4391375266","https://openalex.org/W1994680671","https://openalex.org/W2000283393","https://openalex.org/W2002320543","https://openalex.org/W2061947244","https://openalex.org/W2150232912","https://openalex.org/W2054940838","https://openalex.org/W3106170641","https://openalex.org/W4321855183","https://openalex.org/W1535031934"],"abstract_inverted_index":{"In":[0],"the":[1,26,37,47,53,83,121,146],"realm":[2],"of":[3,17,23,30,32,36,123,156],"Boltzmann-Gibbs":[4],"(BG)":[5],"statistical":[6],"mechanics":[7],"and":[8,90,103,114],"its":[9],"q-generalisation":[10],"for":[11,148],"complex":[12,125],"systems,":[13],"we":[14,137],"analysed":[15],"sequences":[16],"q-triplets,":[18],"or":[19,132],"q-doublets":[20],"if":[21],"one":[22],"them":[24],"was":[25],"unity,":[27],"in":[28,145],"terms":[29],"cycles":[31],"successive":[33],"M\u00f6bius":[34],"transforms":[35,51],"line":[38],"preserving":[39],"unity":[40],"(":[41,57,67,98],"q":[42,55,59,72,96,101,104,108],"=":[43,87,92],"1":[44,61,68,89,106],"corresponds":[45],"to":[46],"BG":[48],"theory).":[49],"Such":[50],"have":[52],"form":[54],"\u21a6":[56,97,105],"a":[58,63,70,74,78,80,86,91,142],"+":[60,69],"\u2212":[62,73,88,100],")":[64,71,102],"/":[65,107],"[":[66],"]":[75],",":[76,109],"where":[77],"is":[79],"real":[81],"number;":[82],"particular":[84],"cases":[85],"0":[93],"yield,":[94],"respectively,":[95],"2":[99],"currently":[110],"known":[111],"as":[112],"additive":[113],"multiplicative":[115],"dualities.":[116],"This":[117],"approach":[118],"seemingly":[119],"enables":[120],"organisation":[122],"various":[124],"phenomena":[126],"into":[127],"different":[128],"classes,":[129],"named":[130],"N-complete":[131],"incomplete.":[133],"The":[134],"classification":[135],"that":[136],"propose":[138],"here":[139],"hopefully":[140],"constitutes":[141],"useful":[143],"guideline":[144],"search,":[147],"non-BG":[149],"systems":[150],"whenever":[151],"well":[152],"described":[153],"through":[154],"q-indices,":[155],"new":[157],"possibly":[158],"observable":[159],"physical":[160],"properties.":[161]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":2},{"year":2022,"cited_by_count":4},{"year":2021,"cited_by_count":4},{"year":2020,"cited_by_count":2}],"updated_date":"2026-05-29T09:21:14.243279","created_date":"2025-10-10T00:00:00"}
