{"id":"https://openalex.org/W2135582882","doi":"https://doi.org/10.3390/e12051145","title":"Entropy: The Markov Ordering Approach","display_name":"Entropy: The Markov Ordering Approach","publication_year":2010,"publication_date":"2010-05-07","ids":{"openalex":"https://openalex.org/W2135582882","doi":"https://doi.org/10.3390/e12051145","mag":"2135582882"},"language":"en","primary_location":{"id":"doi:10.3390/e12051145","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e12051145","pdf_url":"https://www.mdpi.com/1099-4300/12/5/1145/pdf?version=1424784735","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/1099-4300/12/5/1145/pdf?version=1424784735","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":null,"display_name":"Alexander N. Gorban","orcid":null},"institutions":[{"id":"https://openalex.org/I153648349","display_name":"University of Leicester","ror":"https://ror.org/04h699437","country_code":"GB","type":"education","lineage":["https://openalex.org/I153648349"]}],"countries":["GB"],"is_corresponding":true,"raw_author_name":"Alexander N. Gorban","raw_affiliation_strings":["Department of Mathematics, University of Leicester, Leicester, UK"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Leicester, Leicester, UK","institution_ids":["https://openalex.org/I153648349"]}]},{"author_position":"middle","author":{"id":null,"display_name":"Pavel A. Gorban","orcid":null},"institutions":[{"id":"https://openalex.org/I4853710","display_name":"Siberian Federal University","ror":"https://ror.org/05fw97k56","country_code":"RU","type":"education","lineage":["https://openalex.org/I4853710"]}],"countries":["RU"],"is_corresponding":false,"raw_author_name":"Pavel A. Gorban","raw_affiliation_strings":["Institute of Space and Information Technologies, Siberian Federal University, Krasnoyarsk, Russia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institute of Space and Information Technologies, Siberian Federal University, Krasnoyarsk, Russia","institution_ids":["https://openalex.org/I4853710"]}]},{"author_position":"last","author":{"id":null,"display_name":"George Judge","orcid":null},"institutions":[{"id":"https://openalex.org/I95457486","display_name":"University of California, Berkeley","ror":"https://ror.org/01an7q238","country_code":"US","type":"education","lineage":["https://openalex.org/I95457486"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"George Judge","raw_affiliation_strings":["Department of Resource Economics, University of California, Berkeley, CA, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Resource Economics, University of California, Berkeley, CA, USA","institution_ids":["https://openalex.org/I95457486"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":3,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I153648349"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":5.8863,"has_fulltext":true,"cited_by_count":77,"citation_normalized_percentile":{"value":0.96947497,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":99},"biblio":{"volume":"12","issue":"5","first_page":"1145","last_page":"1193"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12261","display_name":"Statistical Mechanics and Entropy","score":0.34119999408721924,"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":0.34119999408721924,"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/T10621","display_name":"Gene Regulatory Network Analysis","score":0.07769999653100967,"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"}},{"id":"https://openalex.org/T10588","display_name":"Mathematical Dynamics and Fractals","score":0.06469999998807907,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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/markov-chain","display_name":"Markov chain","score":0.7382000088691711},{"id":"https://openalex.org/keywords/monotonic-function","display_name":"Monotonic function","score":0.6840999722480774},{"id":"https://openalex.org/keywords/markov-kernel","display_name":"Markov kernel","score":0.5109000205993652},{"id":"https://openalex.org/keywords/entropy","display_name":"Entropy (arrow of time)","score":0.4438000023365021},{"id":"https://openalex.org/keywords/invariant","display_name":"Invariant (physics)","score":0.42149999737739563},{"id":"https://openalex.org/keywords/markov-process","display_name":"Markov process","score":0.42149999737739563},{"id":"https://openalex.org/keywords/regular-polygon","display_name":"Regular polygon","score":0.42100000381469727},{"id":"https://openalex.org/keywords/markov-property","display_name":"Markov property","score":0.39989998936653137}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.760699987411499},{"id":"https://openalex.org/C98763669","wikidata":"https://www.wikidata.org/wiki/Q176645","display_name":"Markov chain","level":2,"score":0.7382000088691711},{"id":"https://openalex.org/C72169020","wikidata":"https://www.wikidata.org/wiki/Q194404","display_name":"Monotonic function","level":2,"score":0.6840999722480774},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5238999724388123},{"id":"https://openalex.org/C106666656","wikidata":"https://www.wikidata.org/wiki/Q6771323","display_name":"Markov kernel","level":5,"score":0.5109000205993652},{"id":"https://openalex.org/C106301342","wikidata":"https://www.wikidata.org/wiki/Q4117933","display_name":"Entropy (arrow of time)","level":2,"score":0.4438000023365021},{"id":"https://openalex.org/C159886148","wikidata":"https://www.wikidata.org/wiki/Q176645","display_name":"Markov process","level":2,"score":0.42149999737739563},{"id":"https://openalex.org/C190470478","wikidata":"https://www.wikidata.org/wiki/Q2370229","display_name":"Invariant (physics)","level":2,"score":0.42149999737739563},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.42100000381469727},{"id":"https://openalex.org/C189973286","wikidata":"https://www.wikidata.org/wiki/Q176695","display_name":"Markov property","level":4,"score":0.39989998936653137},{"id":"https://openalex.org/C204241405","wikidata":"https://www.wikidata.org/wiki/Q461499","display_name":"Transformation (genetics)","level":3,"score":0.35600000619888306},{"id":"https://openalex.org/C163836022","wikidata":"https://www.wikidata.org/wiki/Q6771326","display_name":"Markov model","level":3,"score":0.3513000011444092},{"id":"https://openalex.org/C54907487","wikidata":"https://www.wikidata.org/wiki/Q7915688","display_name":"Variable-order Markov model","level":4,"score":0.34709998965263367},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.33059999346733093},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.32659998536109924},{"id":"https://openalex.org/C18648836","wikidata":"https://www.wikidata.org/wiki/Q381892","display_name":"Compact space","level":2,"score":0.3122999966144562},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3100000023841858},{"id":"https://openalex.org/C60640748","wikidata":"https://www.wikidata.org/wiki/Q2337858","display_name":"Lyapunov function","level":3,"score":0.2985000014305115},{"id":"https://openalex.org/C163540672","wikidata":"https://www.wikidata.org/wiki/Q1115431","display_name":"Markov renewal process","level":5,"score":0.2809999883174896},{"id":"https://openalex.org/C98735484","wikidata":"https://www.wikidata.org/wiki/Q186971","display_name":"Time reversibility","level":5,"score":0.25459998846054077},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.25440001487731934}],"mesh":[],"locations_count":5,"locations":[{"id":"doi:10.3390/e12051145","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e12051145","pdf_url":"https://www.mdpi.com/1099-4300/12/5/1145/pdf?version=1424784735","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:1003.1377","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1003.1377","pdf_url":"https://arxiv.org/pdf/1003.1377","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:doaj.org/article:277e878bd4004039b272868ea0fa7008","is_oa":true,"landing_page_url":"https://doaj.org/article/277e878bd4004039b272868ea0fa7008","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 12, Iss 5, Pp 1145-1193 (2010)","raw_type":"article"},{"id":"pmh:oai:lra.le.ac.uk:2381/20396","is_oa":true,"landing_page_url":"http://www.mdpi.com/1099-4300/12/5/1145","pdf_url":null,"source":{"id":"https://openalex.org/S4306402365","display_name":"Leicester Research Archive (University of Leicester)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I153648349","host_organization_name":"University of Leicester","host_organization_lineage":["https://openalex.org/I153648349"],"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":"","raw_type":"Journal Article"},{"id":"pmh:oai:mdpi.com:/1099-4300/12/5/1145/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/e12051145","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 12; Issue 5; Pages: 1145-1193","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/e12051145","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e12051145","pdf_url":"https://www.mdpi.com/1099-4300/12/5/1145/pdf?version=1424784735","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/W2135582882.pdf","grobid_xml":"https://content.openalex.org/works/W2135582882.grobid-xml"},"referenced_works_count":62,"referenced_works":["https://openalex.org/W642585845","https://openalex.org/W1516900628","https://openalex.org/W1545744341","https://openalex.org/W1564500292","https://openalex.org/W1606879187","https://openalex.org/W1623903145","https://openalex.org/W1965167470","https://openalex.org/W1965555277","https://openalex.org/W1968527826","https://openalex.org/W1971934719","https://openalex.org/W1972639576","https://openalex.org/W1981380156","https://openalex.org/W1982244066","https://openalex.org/W1983722334","https://openalex.org/W1983874169","https://openalex.org/W1985630476","https://openalex.org/W1987138890","https://openalex.org/W1991879533","https://openalex.org/W1995875735","https://openalex.org/W2003068273","https://openalex.org/W2006760715","https://openalex.org/W2021061206","https://openalex.org/W2021840054","https://openalex.org/W2032558547","https://openalex.org/W2033468335","https://openalex.org/W2037027724","https://openalex.org/W2041415618","https://openalex.org/W2043691077","https://openalex.org/W2047513944","https://openalex.org/W2050249319","https://openalex.org/W2051604243","https://openalex.org/W2051657238","https://openalex.org/W2053047297","https://openalex.org/W2055281720","https://openalex.org/W2057495405","https://openalex.org/W2059889790","https://openalex.org/W2060696537","https://openalex.org/W2062700386","https://openalex.org/W2066400352","https://openalex.org/W2068728923","https://openalex.org/W2070945723","https://openalex.org/W2072354677","https://openalex.org/W2072837915","https://openalex.org/W2074412592","https://openalex.org/W2081231767","https://openalex.org/W2082258471","https://openalex.org/W2089954718","https://openalex.org/W2126066275","https://openalex.org/W2165469059","https://openalex.org/W2167191107","https://openalex.org/W2169495281","https://openalex.org/W2490509919","https://openalex.org/W2584365726","https://openalex.org/W3021128674","https://openalex.org/W4230969717","https://openalex.org/W4249667877","https://openalex.org/W4252028749","https://openalex.org/W4252816148","https://openalex.org/W4294841944","https://openalex.org/W4301140016","https://openalex.org/W6600342505","https://openalex.org/W6606269234"],"related_works":[],"abstract_inverted_index":{"The":[0,109],"focus":[1],"of":[2,15,33,47,53,67,70,92,120,133],"this":[3,125],"article":[4],"is":[5],"on":[6],"entropy":[7,121],"and":[8,25,50,82],"Markov":[9,76,102,107],"processes.":[10],"We":[11,86],"study":[12],"the":[13,39,45,59,65,68,88,93,114,118],"properties":[14,80],"functionals":[16,74],"which":[17,37,57,78,99],"are":[18,84,104],"invariant":[19,28],"with":[20,42,62,96],"respect":[21,43,63,97],"to":[22,44,64,98],"monotonic":[23,35,55,105],"transformations":[24],"analyze":[26],"two":[27],"\u201cadditivity\u201d":[29],"properties:":[30],"(i)":[31,81],"existence":[32,52],"a":[34,54,129],"transformation":[36,56],"makes":[38,58],"functional":[40,60],"additive":[41,61],"joining":[46],"independent":[48],"systems":[49],"(ii)":[51,83],"partitioning":[66],"space":[69],"states.":[71],"All":[72],"Lyapunov":[73],"for":[75],"chains":[77],"have":[79],"derived.":[85],"describe":[87],"most":[89],"general":[90],"ordering":[91,115],"distribution":[94],"space,":[95],"all":[100],"continuous-time":[101],"processes":[103],"(the":[106],"order).":[108],"solution":[110],"differs":[111],"significantly":[112],"from":[113],"given":[116],"by":[117],"inequality":[119],"growth.":[122],"For":[123],"inference,":[124],"approach":[126],"results":[127],"in":[128],"convex":[130],"compact":[131],"set":[132],"conditionally":[134],"\u201cmost":[135],"random\u201d":[136],"distributions.":[137]},"counts_by_year":[{"year":2024,"cited_by_count":4},{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":3},{"year":2020,"cited_by_count":2},{"year":2019,"cited_by_count":8},{"year":2018,"cited_by_count":9},{"year":2017,"cited_by_count":1},{"year":2016,"cited_by_count":8},{"year":2015,"cited_by_count":10},{"year":2014,"cited_by_count":4},{"year":2013,"cited_by_count":8},{"year":2012,"cited_by_count":7}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2016-06-24T00:00:00"}