{"id":"https://openalex.org/W1980977025","doi":"https://doi.org/10.3390/e15082989","title":"Time Evolution of Relative Entropies for Anomalous Diffusion","display_name":"Time Evolution of Relative Entropies for Anomalous Diffusion","publication_year":2013,"publication_date":"2013-07-26","ids":{"openalex":"https://openalex.org/W1980977025","doi":"https://doi.org/10.3390/e15082989","mag":"1980977025"},"language":"en","primary_location":{"id":"doi:10.3390/e15082989","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e15082989","pdf_url":"https://www.mdpi.com/1099-4300/15/8/2989/pdf?version=1424785031","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"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/1099-4300/15/8/2989/pdf?version=1424785031","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5052033297","display_name":"Janett Prehl","orcid":"https://orcid.org/0000-0002-8463-4611"},"institutions":[{"id":"https://openalex.org/I2610724","display_name":"Chemnitz University of Technology","ror":"https://ror.org/00a208s56","country_code":"DE","type":"education","lineage":["https://openalex.org/I2610724"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Janett Prehl","raw_affiliation_strings":["Institut f\u00fcr Physik, Technische Universit\u00e4t Chemnitz, D-09107 Chemnitz, Germany"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institut f\u00fcr Physik, Technische Universit\u00e4t Chemnitz, D-09107 Chemnitz, Germany","institution_ids":["https://openalex.org/I2610724"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5008058605","display_name":"Frank Boldt","orcid":null},"institutions":[{"id":"https://openalex.org/I2610724","display_name":"Chemnitz University of Technology","ror":"https://ror.org/00a208s56","country_code":"DE","type":"education","lineage":["https://openalex.org/I2610724"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Frank Boldt","raw_affiliation_strings":["Institut f\u00fcr Physik, Technische Universit\u00e4t Chemnitz, D-09107 Chemnitz, Germany"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institut f\u00fcr Physik, Technische Universit\u00e4t Chemnitz, D-09107 Chemnitz, Germany","institution_ids":["https://openalex.org/I2610724"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5007238192","display_name":"Christopher Essex","orcid":null},"institutions":[{"id":"https://openalex.org/I125749732","display_name":"Western University","ror":"https://ror.org/02grkyz14","country_code":"CA","type":"education","lineage":["https://openalex.org/I125749732"]}],"countries":["CA"],"is_corresponding":false,"raw_author_name":"Christopher Essex","raw_affiliation_strings":["Department of Applied Mathematics, The University of Western Ontario, N6A 5B7 London, Canada"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Applied Mathematics, The University of Western Ontario, N6A 5B7 London, Canada","institution_ids":["https://openalex.org/I125749732"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5111759131","display_name":"Karl Heinz Hoffmann","orcid":null},"institutions":[{"id":"https://openalex.org/I2610724","display_name":"Chemnitz University of Technology","ror":"https://ror.org/00a208s56","country_code":"DE","type":"education","lineage":["https://openalex.org/I2610724"]}],"countries":["DE"],"is_corresponding":true,"raw_author_name":"Karl Hoffmann","raw_affiliation_strings":["Institut f\u00fcr Physik, Technische Universit\u00e4t Chemnitz, D-09107 Chemnitz, Germany"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institut f\u00fcr Physik, Technische Universit\u00e4t Chemnitz, D-09107 Chemnitz, Germany","institution_ids":["https://openalex.org/I2610724"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5111759131"],"corresponding_institution_ids":["https://openalex.org/I2610724"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2227},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2227},"fwci":1.984,"has_fulltext":true,"cited_by_count":16,"citation_normalized_percentile":{"value":0.83902047,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"15","issue":"8","first_page":"2989","last_page":"3006"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10288","display_name":"Fractional Differential Equations Solutions","score":0.9997000098228455,"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"}},"topics":[{"id":"https://openalex.org/T10288","display_name":"Fractional Differential Equations Solutions","score":0.9997000098228455,"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"}},{"id":"https://openalex.org/T12261","display_name":"Statistical Mechanics and Entropy","score":0.9976999759674072,"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/T11513","display_name":"stochastic dynamics and bifurcation","score":0.9872000217437744,"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/entropy-production","display_name":"Entropy production","score":0.6810050010681152},{"id":"https://openalex.org/keywords/statistical-physics","display_name":"Statistical physics","score":0.6790202856063843},{"id":"https://openalex.org/keywords/anomalous-diffusion","display_name":"Anomalous diffusion","score":0.5868583917617798},{"id":"https://openalex.org/keywords/diffusion-process","display_name":"Diffusion process","score":0.5232849717140198},{"id":"https://openalex.org/keywords/entropy","display_name":"Entropy (arrow of time)","score":0.5191868543624878},{"id":"https://openalex.org/keywords/diffusion","display_name":"Diffusion","score":0.45239371061325073},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.44920340180397034},{"id":"https://openalex.org/keywords/spacetime","display_name":"Spacetime","score":0.4375622570514679},{"id":"https://openalex.org/keywords/kullback\u2013leibler-divergence","display_name":"Kullback\u2013Leibler divergence","score":0.435314416885376},{"id":"https://openalex.org/keywords/thermodynamics","display_name":"Thermodynamics","score":0.36986857652664185},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.3661702275276184},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.1785849928855896},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.13119974732398987},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.07416972517967224},{"id":"https://openalex.org/keywords/innovation-diffusion","display_name":"Innovation diffusion","score":0.07289722561836243}],"concepts":[{"id":"https://openalex.org/C207282930","wikidata":"https://www.wikidata.org/wiki/Q5380811","display_name":"Entropy production","level":2,"score":0.6810050010681152},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.6790202856063843},{"id":"https://openalex.org/C164602753","wikidata":"https://www.wikidata.org/wiki/Q567544","display_name":"Anomalous diffusion","level":3,"score":0.5868583917617798},{"id":"https://openalex.org/C68710425","wikidata":"https://www.wikidata.org/wiki/Q5275442","display_name":"Diffusion process","level":3,"score":0.5232849717140198},{"id":"https://openalex.org/C106301342","wikidata":"https://www.wikidata.org/wiki/Q4117933","display_name":"Entropy (arrow of time)","level":2,"score":0.5191868543624878},{"id":"https://openalex.org/C69357855","wikidata":"https://www.wikidata.org/wiki/Q163214","display_name":"Diffusion","level":2,"score":0.45239371061325073},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.44920340180397034},{"id":"https://openalex.org/C130187892","wikidata":"https://www.wikidata.org/wiki/Q133327","display_name":"Spacetime","level":2,"score":0.4375622570514679},{"id":"https://openalex.org/C171752962","wikidata":"https://www.wikidata.org/wiki/Q255166","display_name":"Kullback\u2013Leibler divergence","level":2,"score":0.435314416885376},{"id":"https://openalex.org/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","level":1,"score":0.36986857652664185},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.3661702275276184},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.1785849928855896},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.13119974732398987},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.07416972517967224},{"id":"https://openalex.org/C3017618536","wikidata":"https://www.wikidata.org/wiki/Q304994","display_name":"Innovation diffusion","level":2,"score":0.07289722561836243},{"id":"https://openalex.org/C56739046","wikidata":"https://www.wikidata.org/wiki/Q192060","display_name":"Knowledge management","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/e15082989","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e15082989","pdf_url":"https://www.mdpi.com/1099-4300/15/8/2989/pdf?version=1424785031","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:doaj.org/article:e4cd46db148a4187bbb63cf4afd1f08c","is_oa":true,"landing_page_url":"https://doaj.org/article/e4cd46db148a4187bbb63cf4afd1f08c","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 8, Pp 2989-3006 (2013)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/1099-4300/15/8/2989/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/e15082989","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 8; Pages: 2989-3006","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/e15082989","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e15082989","pdf_url":"https://www.mdpi.com/1099-4300/15/8/2989/pdf?version=1424785031","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":[{"id":"https://openalex.org/F4320320879","display_name":"Deutsche Forschungsgemeinschaft","ror":"https://ror.org/018mejw64"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W1980977025.pdf","grobid_xml":"https://content.openalex.org/works/W1980977025.grobid-xml"},"referenced_works_count":47,"referenced_works":["https://openalex.org/W145038029","https://openalex.org/W379204093","https://openalex.org/W433794350","https://openalex.org/W1674254454","https://openalex.org/W1964583879","https://openalex.org/W1964805502","https://openalex.org/W1965555277","https://openalex.org/W1969547415","https://openalex.org/W1973950856","https://openalex.org/W1976776824","https://openalex.org/W1978541827","https://openalex.org/W1980266537","https://openalex.org/W1992631890","https://openalex.org/W2000593856","https://openalex.org/W2011941390","https://openalex.org/W2012021517","https://openalex.org/W2013012370","https://openalex.org/W2013855261","https://openalex.org/W2022352949","https://openalex.org/W2026305126","https://openalex.org/W2027790217","https://openalex.org/W2035079509","https://openalex.org/W2035632748","https://openalex.org/W2036774377","https://openalex.org/W2038513688","https://openalex.org/W2038857613","https://openalex.org/W2042977647","https://openalex.org/W2044322567","https://openalex.org/W2049361783","https://openalex.org/W2061878175","https://openalex.org/W2062185997","https://openalex.org/W2079252694","https://openalex.org/W2082189977","https://openalex.org/W2087337573","https://openalex.org/W2089045143","https://openalex.org/W2097711273","https://openalex.org/W2100376401","https://openalex.org/W2109722474","https://openalex.org/W2111271983","https://openalex.org/W2124470848","https://openalex.org/W2125317411","https://openalex.org/W2485577080","https://openalex.org/W2552207324","https://openalex.org/W2762994171","https://openalex.org/W3098903841","https://openalex.org/W4211196861","https://openalex.org/W4231695981"],"related_works":["https://openalex.org/W1996593003","https://openalex.org/W2964190585","https://openalex.org/W4390026245","https://openalex.org/W2803757496","https://openalex.org/W1618612856","https://openalex.org/W2379959577","https://openalex.org/W2026946703","https://openalex.org/W4327756654","https://openalex.org/W4318606473","https://openalex.org/W2032935596"],"abstract_inverted_index":{"The":[0],"entropy":[1,25,44],"production":[2,26],"paradox":[3],"for":[4],"anomalous":[5],"diffusion":[6,20,52,88],"processes":[7],"describes":[8],"a":[9,76],"phenomenon":[10],"where":[11],"one-parameter":[12,117],"families":[13],"of":[14,80,106,114,119],"dynamical":[15,120],"equations,":[16,23],"falling":[17],"between":[18,51],"the":[19,35,43,48,103,107,111,115],"and":[21,53],"wave":[22,36,93],"have":[24],"rates":[27],"(Shannon,":[28],"Tsallis":[29],"or":[30],"Renyi)":[31],"that":[32,62],"increase":[33],"toward":[34],"equation":[37],"limit":[38],"unexpectedly.":[39],"Moreover,":[40],"also":[41],"surprisingly,":[42],"does":[45],"not":[46],"order":[47],"bridging":[49],"regime":[50],"waves":[54],"at":[55],"all.":[56],"However,":[57],"it":[58],"has":[59],"been":[60],"found":[61],"relative":[63,108],"entropies,":[64,73],"with":[65],"an":[66],"appropriately":[67],"chosen":[68],"reference":[69],"distribution,":[70],"do.":[71],"Relative":[72],"thus,":[74],"provide":[75],"physically":[77],"sensible":[78],"way":[79],"setting":[81],"which":[82],"process":[83],"is":[84],"\u201cnearer\u201d":[85],"to":[86],"pure":[87,92,98],"than":[89],"another,":[90],"placing":[91],"propagation,":[94],"desirably,":[95],"\u201cfurthest\u201d":[96],"from":[97],"diffusion.":[99],"We":[100],"examine":[101],"here":[102],"time":[104],"behavior":[105],"entropies":[109],"under":[110],"evolution":[112],"dynamics":[113],"underlying":[116],"family":[118],"equations":[121],"based":[122],"on":[123],"space-fractional":[124],"derivatives.":[125]},"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":1},{"year":2020,"cited_by_count":1},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":1},{"year":2016,"cited_by_count":2},{"year":2015,"cited_by_count":2},{"year":2014,"cited_by_count":2},{"year":2013,"cited_by_count":2}],"updated_date":"2026-08-05T07:39:15.569665","created_date":"2025-10-10T00:00:00"}