{"id":"https://openalex.org/W2056215740","doi":"https://doi.org/10.3390/e15125324","title":"Generalized (c,d)-Entropy and Aging Random Walks","display_name":"Generalized (c,d)-Entropy and Aging Random Walks","publication_year":2013,"publication_date":"2013-12-03","ids":{"openalex":"https://openalex.org/W2056215740","doi":"https://doi.org/10.3390/e15125324","mag":"2056215740"},"language":"en","primary_location":{"id":"doi:10.3390/e15125324","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e15125324","pdf_url":"https://www.mdpi.com/1099-4300/15/12/5324/pdf?version=1424785188","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/1099-4300/15/12/5324/pdf?version=1424785188","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5019906371","display_name":"Rudolf Hanel","orcid":"https://orcid.org/0000-0002-4045-9532"},"institutions":[{"id":"https://openalex.org/I76134821","display_name":"Medical University of Vienna","ror":"https://ror.org/05n3x4p02","country_code":"AT","type":"education","lineage":["https://openalex.org/I76134821"]}],"countries":["AT"],"is_corresponding":true,"raw_author_name":"Rudolf Hanel","raw_affiliation_strings":["Section for Science of Complex Systems, Medical University of Vienna, Spitalgasse 23, Vienna A-1090, Austria"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Section for Science of Complex Systems, Medical University of Vienna, Spitalgasse 23, Vienna A-1090, Austria","institution_ids":["https://openalex.org/I76134821"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5045565551","display_name":"Stefan Thurner","orcid":"https://orcid.org/0000-0003-2495-6819"},"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/I1317774081","display_name":"International Institute for Applied Systems Analysis","ror":"https://ror.org/02wfhk785","country_code":"AT","type":"nonprofit","lineage":["https://openalex.org/I1317774081"]},{"id":"https://openalex.org/I76134821","display_name":"Medical University of Vienna","ror":"https://ror.org/05n3x4p02","country_code":"AT","type":"education","lineage":["https://openalex.org/I76134821"]}],"countries":["AT","US"],"is_corresponding":true,"raw_author_name":"Stefan Thurner","raw_affiliation_strings":["Institute for Applied Systems Analysis, Schlossplatz 1, Laxenburg A-2361, Austria","Santa Fe Institute,1399 Hyde Park Road, Santa Fe, NM87501, USA","Section for Science of Complex Systems, Medical University of Vienna, Spitalgasse 23, Vienna A-1090, Austria"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institute for Applied Systems Analysis, Schlossplatz 1, Laxenburg A-2361, Austria","institution_ids":["https://openalex.org/I1317774081"]},{"raw_affiliation_string":"Santa Fe Institute,1399 Hyde Park Road, Santa Fe, NM87501, USA","institution_ids":["https://openalex.org/I1308548392"]},{"raw_affiliation_string":"Section for Science of Complex Systems, Medical University of Vienna, Spitalgasse 23, Vienna A-1090, Austria","institution_ids":["https://openalex.org/I76134821"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5019906371","https://openalex.org/A5045565551"],"corresponding_institution_ids":["https://openalex.org/I1308548392","https://openalex.org/I1317774081","https://openalex.org/I76134821"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2227},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2227},"fwci":3.2086,"has_fulltext":true,"cited_by_count":36,"citation_normalized_percentile":{"value":0.92088551,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"15","issue":"12","first_page":"5324","last_page":"5337"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12261","display_name":"Statistical Mechanics and Entropy","score":0.9997000098228455,"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":0.9997000098228455,"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":0.9703999757766724,"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/T11520","display_name":"Advanced Thermodynamics and Statistical Mechanics","score":0.9674000144004822,"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/ergodic-theory","display_name":"Ergodic theory","score":0.7505097389221191},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6707116961479187},{"id":"https://openalex.org/keywords/statistical-physics","display_name":"Statistical physics","score":0.6493610143661499},{"id":"https://openalex.org/keywords/random-walk","display_name":"Random walk","score":0.629431426525116},{"id":"https://openalex.org/keywords/entropy","display_name":"Entropy (arrow of time)","score":0.5664333701133728},{"id":"https://openalex.org/keywords/markov-process","display_name":"Markov process","score":0.5178568363189697},{"id":"https://openalex.org/keywords/tsallis-entropy","display_name":"Tsallis entropy","score":0.46962809562683105},{"id":"https://openalex.org/keywords/r\u00e9nyi-entropy","display_name":"R\u00e9nyi entropy","score":0.468131422996521},{"id":"https://openalex.org/keywords/scaling","display_name":"Scaling","score":0.42436400055885315},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.40829336643218994},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.36033913493156433},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.2128855288028717},{"id":"https://openalex.org/keywords/principle-of-maximum-entropy","display_name":"Principle of maximum entropy","score":0.18626469373703003},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.1566966474056244},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.08743751049041748},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.08447203040122986}],"concepts":[{"id":"https://openalex.org/C122044880","wikidata":"https://www.wikidata.org/wiki/Q5498822","display_name":"Ergodic theory","level":2,"score":0.7505097389221191},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6707116961479187},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.6493610143661499},{"id":"https://openalex.org/C121194460","wikidata":"https://www.wikidata.org/wiki/Q856741","display_name":"Random walk","level":2,"score":0.629431426525116},{"id":"https://openalex.org/C106301342","wikidata":"https://www.wikidata.org/wiki/Q4117933","display_name":"Entropy (arrow of time)","level":2,"score":0.5664333701133728},{"id":"https://openalex.org/C159886148","wikidata":"https://www.wikidata.org/wiki/Q176645","display_name":"Markov process","level":2,"score":0.5178568363189697},{"id":"https://openalex.org/C117521176","wikidata":"https://www.wikidata.org/wiki/Q7849341","display_name":"Tsallis entropy","level":3,"score":0.46962809562683105},{"id":"https://openalex.org/C142611142","wikidata":"https://www.wikidata.org/wiki/Q1433083","display_name":"R\u00e9nyi entropy","level":3,"score":0.468131422996521},{"id":"https://openalex.org/C99844830","wikidata":"https://www.wikidata.org/wiki/Q102441924","display_name":"Scaling","level":2,"score":0.42436400055885315},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.40829336643218994},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.36033913493156433},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.2128855288028717},{"id":"https://openalex.org/C9679016","wikidata":"https://www.wikidata.org/wiki/Q1417473","display_name":"Principle of maximum entropy","level":2,"score":0.18626469373703003},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.1566966474056244},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.08743751049041748},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.08447203040122986},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0}],"mesh":[],"locations_count":5,"locations":[{"id":"doi:10.3390/e15125324","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e15125324","pdf_url":"https://www.mdpi.com/1099-4300/15/12/5324/pdf?version=1424785188","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:1310.5959","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1310.5959","pdf_url":"https://arxiv.org/pdf/1310.5959","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"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:pure.iiasa.ac.at:10306","is_oa":true,"landing_page_url":"http://pure.iiasa.ac.at/id/eprint/10306/1/Generalized%20%28c%2Cd%29-Entropy%20and%20Aging%20Random%20Walks.pdf","pdf_url":"http://pure.iiasa.ac.at/id/eprint/10306/1/Generalized%20%28c%2Cd%29-Entropy%20and%20Aging%20Random%20Walks.pdf","source":{"id":"https://openalex.org/S4306401423","display_name":"IIASA PURE (International Institute of Applied Systems Analysis)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1317774081","host_organization_name":"International Institute for Applied Systems Analysis","host_organization_lineage":["https://openalex.org/I1317774081"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Article"},{"id":"pmh:oai:doaj.org/article:958de9c6ee5843129fa4336c662950c9","is_oa":true,"landing_page_url":"https://doaj.org/article/958de9c6ee5843129fa4336c662950c9","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 15, Iss 12, Pp 5324-5337 (2013)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/1099-4300/15/12/5324/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/e15125324","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 15; Issue 12; Pages: 5324-5337","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/e15125324","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e15125324","pdf_url":"https://www.mdpi.com/1099-4300/15/12/5324/pdf?version=1424785188","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":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2056215740.pdf","grobid_xml":"https://content.openalex.org/works/W2056215740.grobid-xml"},"referenced_works_count":28,"referenced_works":["https://openalex.org/W1511888687","https://openalex.org/W1672879477","https://openalex.org/W1848147104","https://openalex.org/W1971582529","https://openalex.org/W1983874169","https://openalex.org/W1989308357","https://openalex.org/W1995875735","https://openalex.org/W1997414622","https://openalex.org/W2000356308","https://openalex.org/W2005074558","https://openalex.org/W2015067433","https://openalex.org/W2021252433","https://openalex.org/W2033491666","https://openalex.org/W2040164803","https://openalex.org/W2057395226","https://openalex.org/W2079584573","https://openalex.org/W2087171258","https://openalex.org/W2088048184","https://openalex.org/W2093988651","https://openalex.org/W2105771700","https://openalex.org/W2136222558","https://openalex.org/W2913075478","https://openalex.org/W2964213744","https://openalex.org/W2993383518","https://openalex.org/W3099497943","https://openalex.org/W3104822214","https://openalex.org/W3148089400","https://openalex.org/W4318580647"],"related_works":["https://openalex.org/W2259764508","https://openalex.org/W4386898430","https://openalex.org/W2127148427","https://openalex.org/W2000533339","https://openalex.org/W2883631711","https://openalex.org/W2911404282","https://openalex.org/W1984359518","https://openalex.org/W3213226332","https://openalex.org/W1980936589","https://openalex.org/W3125519806"],"abstract_inverted_index":{"Complex":[0],"systems":[1,37,129,139],"are":[2,138,145],"often":[3],"inherently":[4],"non-ergodic":[5,25,36],"and":[6,8,16,26,48,73,81,96,147],"non-Markovian":[7,27,112],"Shannon":[9,95],"entropy":[10,34,59,84,89,104],"loses":[11],"its":[12],"applicability.":[13],"Accelerating,":[14],"path-dependent":[15],"aging":[17,154],"random":[18,113],"walks":[19,121],"offer":[20],"an":[21],"intuitive":[22],"picture":[23],"for":[24,107,152],"systems.":[28],"It":[29,99],"was":[30,100],"shown":[31,101],"that":[32,102,130,151],"the":[33,45,51,53,62,159,164,168,172],"of":[35,44,61,111,127,167,174],"can":[38,131],"still":[39],"be":[40,132],"derived":[41],"from":[42],"three":[43],"Shannon\u2013Khinchin":[46],"axioms":[47],"by":[49],"violating":[50],"fourth,":[52],"so-called":[54],"composition":[55],"axiom.":[56],"The":[57],"corresponding":[58],"is":[60,105,157],"form":[63],"Sc,d":[64,156],"~":[65],"\u2211i\u0393(1":[66],"+":[67],"d,":[68],"1":[69],"\u2212":[70],"cln":[71],"pi)":[72],"depends":[74],"on":[75],"two":[76],"system-specific":[77],"scaling":[78],"exponents,":[79],"c":[80],"d.":[82],"This":[83],"contains":[85],"many":[86],"recently":[87],"proposed":[88],"functionals":[90],"as":[91,134],"special":[92,109],"cases,":[93],"including":[94],"Tsallis":[97],"entropy.":[98,162],"this":[103,116],"relevant":[106],"a":[108,123,177],"class":[110,126],"walks.":[114,136],"In":[115],"work,":[117],"we":[118,170],"generalize":[119],"these":[120],"to":[122],"much":[124],"wider":[125],"stochastic":[128],"characterized":[133],"\u201caging\u201d":[135],"These":[137],"whose":[140],"transition":[141],"rates":[142],"between":[143],"states":[144],"path-":[146],"time-dependent.":[148],"We":[149],"show":[150],"particular":[153],"walks,":[155],"again":[158],"correct":[160],"extensive":[161],"Before":[163],"central":[165],"part":[166],"paper,":[169],"review":[171],"concept":[173],"(c,d)-entropy":[175],"in":[176],"self-contained":[178],"way.":[179]},"counts_by_year":[{"year":2023,"cited_by_count":3},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":2},{"year":2019,"cited_by_count":2},{"year":2018,"cited_by_count":9},{"year":2017,"cited_by_count":4},{"year":2016,"cited_by_count":5},{"year":2015,"cited_by_count":6},{"year":2014,"cited_by_count":3}],"updated_date":"2026-08-11T07:18:39.950985","created_date":"2025-10-10T00:00:00"}
