{"id":"https://openalex.org/W3030453454","doi":"https://doi.org/10.3390/e22060594","title":"Nonextensive Statistical Mechanics: Equivalence Between Dual Entropy and Dual Probabilities","display_name":"Nonextensive Statistical Mechanics: Equivalence Between Dual Entropy and Dual Probabilities","publication_year":2020,"publication_date":"2020-05-26","ids":{"openalex":"https://openalex.org/W3030453454","doi":"https://doi.org/10.3390/e22060594","mag":"3030453454","pmid":"https://pubmed.ncbi.nlm.nih.gov/33286366"},"language":"en","primary_location":{"id":"doi:10.3390/e22060594","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e22060594","pdf_url":"https://www.mdpi.com/1099-4300/22/6/594/pdf?version=1591331740","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","doaj","pubmed"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/1099-4300/22/6/594/pdf?version=1591331740","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5080379161","display_name":"G. Livadiotis","orcid":"https://orcid.org/0000-0002-7655-6019"},"institutions":[{"id":"https://openalex.org/I1290463931","display_name":"Southwest Research Institute","ror":"https://ror.org/03tghng59","country_code":"US","type":"nonprofit","lineage":["https://openalex.org/I1290463931"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"George Livadiotis","raw_affiliation_strings":["Division of Space Science and Engineering, Southwest Research Institute, San Antonio, TX 78238, USA"],"raw_orcid":"https://orcid.org/0000-0002-7655-6019","affiliations":[{"raw_affiliation_string":"Division of Space Science and Engineering, Southwest Research Institute, San Antonio, TX 78238, USA","institution_ids":["https://openalex.org/I1290463931"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5080379161"],"corresponding_institution_ids":["https://openalex.org/I1290463931"],"apc_list":{"value":1600,"currency":"CHF","value_usd":1690},"apc_paid":{"value":1600,"currency":"CHF","value_usd":1690},"fwci":0.1191,"has_fulltext":true,"cited_by_count":2,"citation_normalized_percentile":{"value":0.40222092,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":97},"biblio":{"volume":"22","issue":"6","first_page":"594","last_page":"594"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12261","display_name":"Statistical Mechanics and Entropy","score":0.9998999834060669,"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":0.9998999834060669,"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/T11270","display_name":"Complex Systems and Time Series Analysis","score":0.9728999733924866,"subfield":{"id":"https://openalex.org/subfields/2002","display_name":"Economics and Econometrics"},"field":{"id":"https://openalex.org/fields/20","display_name":"Economics, Econometrics and Finance"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},{"id":"https://openalex.org/T10591","display_name":"Theoretical and Computational Physics","score":0.9581999778747559,"subfield":{"id":"https://openalex.org/subfields/3104","display_name":"Condensed Matter 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/statistical-mechanics","display_name":"Statistical mechanics","score":0.6693053245544434},{"id":"https://openalex.org/keywords/statistical-physics","display_name":"Statistical physics","score":0.6089974641799927},{"id":"https://openalex.org/keywords/dual","display_name":"Dual (grammatical number)","score":0.5922244787216187},{"id":"https://openalex.org/keywords/equivalence","display_name":"Equivalence (formal languages)","score":0.5857475996017456},{"id":"https://openalex.org/keywords/entropy","display_name":"Entropy (arrow of time)","score":0.5282706022262573},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5127495527267456},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.30737102031707764},{"id":"https://openalex.org/keywords/thermodynamics","display_name":"Thermodynamics","score":0.17488771677017212},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.15057826042175293},{"id":"https://openalex.org/keywords/philosophy","display_name":"Philosophy","score":0.09158357977867126}],"concepts":[{"id":"https://openalex.org/C99874945","wikidata":"https://www.wikidata.org/wiki/Q188715","display_name":"Statistical mechanics","level":2,"score":0.6693053245544434},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.6089974641799927},{"id":"https://openalex.org/C2780980858","wikidata":"https://www.wikidata.org/wiki/Q110022","display_name":"Dual (grammatical number)","level":2,"score":0.5922244787216187},{"id":"https://openalex.org/C2780069185","wikidata":"https://www.wikidata.org/wiki/Q7977945","display_name":"Equivalence (formal languages)","level":2,"score":0.5857475996017456},{"id":"https://openalex.org/C106301342","wikidata":"https://www.wikidata.org/wiki/Q4117933","display_name":"Entropy (arrow of time)","level":2,"score":0.5282706022262573},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5127495527267456},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.30737102031707764},{"id":"https://openalex.org/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","level":1,"score":0.17488771677017212},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.15057826042175293},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.09158357977867126},{"id":"https://openalex.org/C41895202","wikidata":"https://www.wikidata.org/wiki/Q8162","display_name":"Linguistics","level":1,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.3390/e22060594","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e22060594","pdf_url":"https://www.mdpi.com/1099-4300/22/6/594/pdf?version=1591331740","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:33286366","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/33286366","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":"Journal Article"},{"id":"pmh:oai:pubmedcentral.nih.gov:7517129","is_oa":true,"landing_page_url":"https://www.ncbi.nlm.nih.gov/pmc/articles/7517129","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":"pmh:oai:doaj.org/article:d3a6439029e54f07b1105ac0d08c0025","is_oa":true,"landing_page_url":"https://doaj.org/article/d3a6439029e54f07b1105ac0d08c0025","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 22, Iss 6, p 594 (2020)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.3390/e22060594","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e22060594","pdf_url":"https://www.mdpi.com/1099-4300/22/6/594/pdf?version=1591331740","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","score":0.7300000190734863,"id":"https://metadata.un.org/sdg/7"}],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W3030453454.pdf","grobid_xml":"https://content.openalex.org/works/W3030453454.grobid-xml"},"referenced_works_count":145,"referenced_works":["https://openalex.org/W1553236419","https://openalex.org/W1553495501","https://openalex.org/W1563520392","https://openalex.org/W1586540887","https://openalex.org/W1592472301","https://openalex.org/W1672879477","https://openalex.org/W1862918737","https://openalex.org/W1914161725","https://openalex.org/W1962503091","https://openalex.org/W1965484822","https://openalex.org/W1966896725","https://openalex.org/W1968296390","https://openalex.org/W1968462845","https://openalex.org/W1969136575","https://openalex.org/W1972138494","https://openalex.org/W1974784750","https://openalex.org/W1982893097","https://openalex.org/W1983874169","https://openalex.org/W1984215176","https://openalex.org/W1986026923","https://openalex.org/W1987723239","https://openalex.org/W1988847757","https://openalex.org/W1991172246","https://openalex.org/W1993089110","https://openalex.org/W1995233652","https://openalex.org/W1997281911","https://openalex.org/W2004359919","https://openalex.org/W2004885945","https://openalex.org/W2007004084","https://openalex.org/W2007180706","https://openalex.org/W2008400532","https://openalex.org/W2009586635","https://openalex.org/W2009678820","https://openalex.org/W2010908969","https://openalex.org/W2012818649","https://openalex.org/W2015351206","https://openalex.org/W2016584282","https://openalex.org/W2017109395","https://openalex.org/W2018454770","https://openalex.org/W2019516386","https://openalex.org/W2021102511","https://openalex.org/W2025461145","https://openalex.org/W2026905733","https://openalex.org/W2027242260","https://openalex.org/W2029527392","https://openalex.org/W2029725976","https://openalex.org/W2031506052","https://openalex.org/W2032077889","https://openalex.org/W2033552170","https://openalex.org/W2033652368","https://openalex.org/W2033751654","https://openalex.org/W2034315815","https://openalex.org/W2034955726","https://openalex.org/W2035637366","https://openalex.org/W2040459036","https://openalex.org/W2043784086","https://openalex.org/W2046583319","https://openalex.org/W2047857569","https://openalex.org/W2050941779","https://openalex.org/W2053333723","https://openalex.org/W2057453691","https://openalex.org/W2057794586","https://openalex.org/W2058172722","https://openalex.org/W2059017926","https://openalex.org/W2061969976","https://openalex.org/W2066334054","https://openalex.org/W2066947923","https://openalex.org/W2068622102","https://openalex.org/W2070847964","https://openalex.org/W2072406744","https://openalex.org/W2074616555","https://openalex.org/W2075193088","https://openalex.org/W2080058987","https://openalex.org/W2080414626","https://openalex.org/W2081523618","https://openalex.org/W2084067829","https://openalex.org/W2086048517","https://openalex.org/W2086722473","https://openalex.org/W2087078284","https://openalex.org/W2087173485","https://openalex.org/W2087617996","https://openalex.org/W2088568410","https://openalex.org/W2089852843","https://openalex.org/W2090170322","https://openalex.org/W2091286941","https://openalex.org/W2095258004","https://openalex.org/W2095258883","https://openalex.org/W2100715189","https://openalex.org/W2109930111","https://openalex.org/W2111164128","https://openalex.org/W2119169243","https://openalex.org/W2121985084","https://openalex.org/W2123696780","https://openalex.org/W2126136850","https://openalex.org/W2127141930","https://openalex.org/W2128987503","https://openalex.org/W2130490793","https://openalex.org/W2133196912","https://openalex.org/W2145587246","https://openalex.org/W2145655997","https://openalex.org/W2147142925","https://openalex.org/W2154003514","https://openalex.org/W2160943512","https://openalex.org/W2163740295","https://openalex.org/W2164500104","https://openalex.org/W2165864744","https://openalex.org/W2170614585","https://openalex.org/W2251803630","https://openalex.org/W2256216980","https://openalex.org/W2277420092","https://openalex.org/W2483389846","https://openalex.org/W2516865766","https://openalex.org/W2563203903","https://openalex.org/W2607345677","https://openalex.org/W2625882416","https://openalex.org/W2759633664","https://openalex.org/W2785731430","https://openalex.org/W2790395485","https://openalex.org/W2884571279","https://openalex.org/W2885749470","https://openalex.org/W2891790758","https://openalex.org/W2895642978","https://openalex.org/W2903730516","https://openalex.org/W2914496275","https://openalex.org/W2953644749","https://openalex.org/W2959420498","https://openalex.org/W2965752544","https://openalex.org/W2983328162","https://openalex.org/W2985716120","https://openalex.org/W3009167218","https://openalex.org/W3099301383","https://openalex.org/W3099538892","https://openalex.org/W3101365170","https://openalex.org/W3101751208","https://openalex.org/W3103119206","https://openalex.org/W3103803711","https://openalex.org/W3104512157","https://openalex.org/W3104515265","https://openalex.org/W3105636028","https://openalex.org/W3106501272","https://openalex.org/W3124259920","https://openalex.org/W4249046827","https://openalex.org/W4298384299","https://openalex.org/W6657751866","https://openalex.org/W6679412393"],"related_works":["https://openalex.org/W1541882734","https://openalex.org/W2317351040","https://openalex.org/W2952466936","https://openalex.org/W4306779659","https://openalex.org/W2307595802","https://openalex.org/W2018846834","https://openalex.org/W3100717469","https://openalex.org/W2091905034","https://openalex.org/W2180033908","https://openalex.org/W2126551324"],"abstract_inverted_index":{"The":[0,30,116],"concept":[1],"of":[2,4,15,20,27,37,55,64,78,95,103,131,134],"duality":[3,32,133],"probability":[5,31,118],"distributions":[6],"constitutes":[7],"a":[8,60,91,96],"fundamental":[9],"\u201cbrick\u201d":[10],"in":[11,52,155,159],"the":[12,25,28,38,43,46,50,53,65,75,101,123,129,132,142,151],"solid":[13],"framework":[14,102],"nonextensive":[16,104],"statistical":[17,22,105],"mechanics\u2014the":[18],"generalization":[19],"Boltzmann\u2013Gibbs":[21],"mechanics":[23,106],"under":[24],"consideration":[26],"q-entropy.":[29],"is":[33,59,139],"solving":[34],"old-standing":[35],"issues":[36],"theory,":[39,66],"e.g.,":[40],"it":[41,58,69,82,138],"ascertains":[42],"additivity":[44,51],"for":[45],"internal":[47],"energy":[48,54],"given":[49],"microstates.":[56],"However,":[57],"rather":[61],"complex":[62],"part":[63],"and":[67,113,146,162],"certainly,":[68],"cannot":[70],"be":[71,108],"trivially":[72],"explained":[73],"along":[74],"Gibb\u2019s":[76],"path":[77],"entropy":[79],"maximization.":[80],"Recently,":[81],"was":[83],"shown":[84,140],"that":[85,141],"an":[86,156],"alternative":[87],"picture":[88],"exists,":[89],"considering":[90],"dual":[92,97,143],"entropy,":[93],"instead":[94],"probability.":[98],"In":[99],"particular,":[100],"can":[107],"equivalently":[109],"developed":[110],"using":[111],"q-":[112],"1/q-":[114],"entropies.":[115],"canonical":[117],"distribution":[119],"coincides":[120],"again":[121],"with":[122],"known":[124],"q-exponential":[125],"distribution,":[126],"but":[127],"without":[128],"necessity":[130],"ordinary-escort":[135],"probabilities.":[136],"Furthermore,":[137],"entropies,":[144],"q-entropy":[145],"1/q-entropy,":[147],"as":[148],"well":[149],"as,":[150],"1-entropy,":[152],"are":[153],"involved":[154],"identity,":[157],"useful":[158],"theoretical":[160],"development":[161],"applications.":[163]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2023,"cited_by_count":1}],"updated_date":"2026-08-28T12:50:07.497085","created_date":"2025-10-10T00:00:00"}