{"id":"https://openalex.org/W4281666366","doi":"https://doi.org/10.3390/e24060802","title":"From Chaos to Ordering: New Studies in the Shannon Entropy of 2D Patterns","display_name":"From Chaos to Ordering: New Studies in the Shannon Entropy of 2D Patterns","publication_year":2022,"publication_date":"2022-06-08","ids":{"openalex":"https://openalex.org/W4281666366","doi":"https://doi.org/10.3390/e24060802","pmid":"https://pubmed.ncbi.nlm.nih.gov/35741523"},"language":"en","primary_location":{"id":"doi:10.3390/e24060802","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e24060802","pdf_url":"https://www.mdpi.com/1099-4300/24/6/802/pdf?version=1654756760","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/1099-4300/24/6/802/pdf?version=1654756760","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5016034099","display_name":"Irina Legchenkova","orcid":"https://orcid.org/0000-0002-6384-7570"},"institutions":[{"id":"https://openalex.org/I52170813","display_name":"Ariel University","ror":"https://ror.org/03nz8qe97","country_code":"IL","type":"education","lineage":["https://openalex.org/I52170813"]}],"countries":["IL"],"is_corresponding":false,"raw_author_name":"Irina Legchenkova","raw_affiliation_strings":["Department of Chemical Engineering, Engineering Faculty, Ariel University, P.O. Box 3, Ariel 407000, Israel"],"raw_orcid":"https://orcid.org/0000-0002-6384-7570","affiliations":[{"raw_affiliation_string":"Department of Chemical Engineering, Engineering Faculty, Ariel University, P.O. Box 3, Ariel 407000, Israel","institution_ids":["https://openalex.org/I52170813"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5034655838","display_name":"Mark Frenkel","orcid":"https://orcid.org/0000-0003-3025-6130"},"institutions":[{"id":"https://openalex.org/I52170813","display_name":"Ariel University","ror":"https://ror.org/03nz8qe97","country_code":"IL","type":"education","lineage":["https://openalex.org/I52170813"]}],"countries":["IL"],"is_corresponding":false,"raw_author_name":"Mark Frenkel","raw_affiliation_strings":["Department of Chemical Engineering, Engineering Faculty, Ariel University, P.O. Box 3, Ariel 407000, Israel"],"raw_orcid":"https://orcid.org/0000-0003-3025-6130","affiliations":[{"raw_affiliation_string":"Department of Chemical Engineering, Engineering Faculty, Ariel University, P.O. Box 3, Ariel 407000, Israel","institution_ids":["https://openalex.org/I52170813"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5079590419","display_name":"Nir Shvalb","orcid":"https://orcid.org/0000-0001-8246-3727"},"institutions":[{"id":"https://openalex.org/I52170813","display_name":"Ariel University","ror":"https://ror.org/03nz8qe97","country_code":"IL","type":"education","lineage":["https://openalex.org/I52170813"]}],"countries":["IL"],"is_corresponding":false,"raw_author_name":"Nir Shvalb","raw_affiliation_strings":["Department of Mechanical Engineering & Mechatronics, Faculty of Engineering, Ariel University, P.O. Box 3, Ariel 407000, Israel"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mechanical Engineering & Mechatronics, Faculty of Engineering, Ariel University, P.O. Box 3, Ariel 407000, Israel","institution_ids":["https://openalex.org/I52170813"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5066005912","display_name":"Shraga Shoval","orcid":"https://orcid.org/0000-0002-0582-4821"},"institutions":[{"id":"https://openalex.org/I52170813","display_name":"Ariel University","ror":"https://ror.org/03nz8qe97","country_code":"IL","type":"education","lineage":["https://openalex.org/I52170813"]}],"countries":["IL"],"is_corresponding":false,"raw_author_name":"Shraga Shoval","raw_affiliation_strings":["Department of Industrial Engineering and Management, Faculty of Engineering, Ariel University, P.O. Box 3, Ariel 407000, Israel"],"raw_orcid":"https://orcid.org/0000-0002-0582-4821","affiliations":[{"raw_affiliation_string":"Department of Industrial Engineering and Management, Faculty of Engineering, Ariel University, P.O. Box 3, Ariel 407000, Israel","institution_ids":["https://openalex.org/I52170813"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5050039013","display_name":"Oleg Gendelman","orcid":"https://orcid.org/0000-0002-4750-2708"},"institutions":[{"id":"https://openalex.org/I174306211","display_name":"Technion \u2013 Israel Institute of Technology","ror":"https://ror.org/03qryx823","country_code":"IL","type":"education","lineage":["https://openalex.org/I174306211"]}],"countries":["IL"],"is_corresponding":false,"raw_author_name":"Oleg V. Gendelman","raw_affiliation_strings":["Faculty of Mechanical Engineering, Technion\u2014Israel Institute of Technology, P.O. Box 10, Haifa 3200003, Israel"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Faculty of Mechanical Engineering, Technion\u2014Israel Institute of Technology, P.O. Box 10, Haifa 3200003, Israel","institution_ids":["https://openalex.org/I174306211"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5088635445","display_name":"Edward Bormashenko","orcid":"https://orcid.org/0000-0003-1356-2486"},"institutions":[{"id":"https://openalex.org/I52170813","display_name":"Ariel University","ror":"https://ror.org/03nz8qe97","country_code":"IL","type":"education","lineage":["https://openalex.org/I52170813"]}],"countries":["IL"],"is_corresponding":true,"raw_author_name":"Edward Bormashenko","raw_affiliation_strings":["Department of Chemical Engineering, Engineering Faculty, Ariel University, P.O. Box 3, Ariel 407000, Israel"],"raw_orcid":"https://orcid.org/0000-0003-1356-2486","affiliations":[{"raw_affiliation_string":"Department of Chemical Engineering, Engineering Faculty, Ariel University, P.O. Box 3, Ariel 407000, Israel","institution_ids":["https://openalex.org/I52170813"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5088635445"],"corresponding_institution_ids":["https://openalex.org/I52170813"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":0.2053,"has_fulltext":false,"cited_by_count":7,"citation_normalized_percentile":{"value":0.35396408,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"24","issue":"6","first_page":"802","last_page":"802"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10474","display_name":"Material Dynamics and Properties","score":0.972599983215332,"subfield":{"id":"https://openalex.org/subfields/2505","display_name":"Materials Chemistry"},"field":{"id":"https://openalex.org/fields/25","display_name":"Materials Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10474","display_name":"Material Dynamics and Properties","score":0.972599983215332,"subfield":{"id":"https://openalex.org/subfields/2505","display_name":"Materials Chemistry"},"field":{"id":"https://openalex.org/fields/25","display_name":"Materials Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10591","display_name":"Theoretical and Computational Physics","score":0.95169997215271,"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"}},{"id":"https://openalex.org/T13126","display_name":"Scientific Research and Discoveries","score":0.9503999948501587,"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/voronoi-diagram","display_name":"Voronoi diagram","score":0.9531424045562744},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7132412791252136},{"id":"https://openalex.org/keywords/entropy","display_name":"Entropy (arrow of time)","score":0.6003249883651733},{"id":"https://openalex.org/keywords/r\u00e9nyi-entropy","display_name":"R\u00e9nyi entropy","score":0.5304571986198425},{"id":"https://openalex.org/keywords/statistical-physics","display_name":"Statistical physics","score":0.4724728763103485},{"id":"https://openalex.org/keywords/centroidal-voronoi-tessellation","display_name":"Centroidal Voronoi tessellation","score":0.4157271981239319},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.3829807937145233},{"id":"https://openalex.org/keywords/principle-of-maximum-entropy","display_name":"Principle of maximum entropy","score":0.2695784568786621},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.2315799593925476},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.15664851665496826},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.13350927829742432},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.0893089771270752}],"concepts":[{"id":"https://openalex.org/C24881265","wikidata":"https://www.wikidata.org/wiki/Q757267","display_name":"Voronoi diagram","level":2,"score":0.9531424045562744},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7132412791252136},{"id":"https://openalex.org/C106301342","wikidata":"https://www.wikidata.org/wiki/Q4117933","display_name":"Entropy (arrow of time)","level":2,"score":0.6003249883651733},{"id":"https://openalex.org/C142611142","wikidata":"https://www.wikidata.org/wiki/Q1433083","display_name":"R\u00e9nyi entropy","level":3,"score":0.5304571986198425},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.4724728763103485},{"id":"https://openalex.org/C205672865","wikidata":"https://www.wikidata.org/wiki/Q5062961","display_name":"Centroidal Voronoi tessellation","level":3,"score":0.4157271981239319},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.3829807937145233},{"id":"https://openalex.org/C9679016","wikidata":"https://www.wikidata.org/wiki/Q1417473","display_name":"Principle of maximum entropy","level":2,"score":0.2695784568786621},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.2315799593925476},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.15664851665496826},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.13350927829742432},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0893089771270752}],"mesh":[],"locations_count":5,"locations":[{"id":"doi:10.3390/e24060802","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e24060802","pdf_url":"https://www.mdpi.com/1099-4300/24/6/802/pdf?version=1654756760","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:35741523","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/35741523","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":null},{"id":"pmh:oai:doaj.org/article:795529759c6e46a3a99707ffdbee185c","is_oa":true,"landing_page_url":"https://doaj.org/article/795529759c6e46a3a99707ffdbee185c","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 24, Iss 6, p 802 (2022)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/1099-4300/24/6/802/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/e24060802","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:9222286","is_oa":true,"landing_page_url":"https://www.ncbi.nlm.nih.gov/pmc/articles/9222286","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/e24060802","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e24060802","pdf_url":"https://www.mdpi.com/1099-4300/24/6/802/pdf?version=1654756760","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":false,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4281666366.pdf"},"referenced_works_count":45,"referenced_works":["https://openalex.org/W138638707","https://openalex.org/W1520127302","https://openalex.org/W1981424957","https://openalex.org/W1989756836","https://openalex.org/W1996537009","https://openalex.org/W2007551675","https://openalex.org/W2008219567","https://openalex.org/W2035903170","https://openalex.org/W2064182541","https://openalex.org/W2064791680","https://openalex.org/W2073622316","https://openalex.org/W2074670256","https://openalex.org/W2090238112","https://openalex.org/W2124579751","https://openalex.org/W2125566231","https://openalex.org/W2135255496","https://openalex.org/W2172264954","https://openalex.org/W2291903977","https://openalex.org/W2332391986","https://openalex.org/W2333612946","https://openalex.org/W2623750799","https://openalex.org/W2781103612","https://openalex.org/W2782572043","https://openalex.org/W2790435797","https://openalex.org/W2883187987","https://openalex.org/W2902610165","https://openalex.org/W2968923792","https://openalex.org/W3006967304","https://openalex.org/W3026676619","https://openalex.org/W3037085755","https://openalex.org/W3044642358","https://openalex.org/W3087591754","https://openalex.org/W3093693011","https://openalex.org/W3095611507","https://openalex.org/W3121259400","https://openalex.org/W3123219654","https://openalex.org/W3123492901","https://openalex.org/W3128013792","https://openalex.org/W3145655529","https://openalex.org/W3197151903","https://openalex.org/W3201444856","https://openalex.org/W3208324164","https://openalex.org/W4220693207","https://openalex.org/W6788999501","https://openalex.org/W6790495338"],"related_works":["https://openalex.org/W2059217232","https://openalex.org/W1992160534","https://openalex.org/W2371724110","https://openalex.org/W1930867958","https://openalex.org/W2053576657","https://openalex.org/W1995318580","https://openalex.org/W4206423751","https://openalex.org/W2041214247","https://openalex.org/W2133344964","https://openalex.org/W2051752778"],"abstract_inverted_index":{"Properties":[0],"of":[1,26,39,67,71,75,78,100,106,156,158,178,188,194,196,214,229],"the":[2,19,31,37,55,68,76,118,121,139,146,153,159,162,168,173,176,186,191,197,201,217,226,230],"Voronoi":[3,20,40,57,72],"tessellations":[4],"arising":[5],"from":[6],"random":[7,45,104,136],"2D":[8,218],"distribution":[9],"points":[10,27,49],"are":[11,132],"reported.":[12],"We":[13],"applied":[14],"an":[15,113,211],"iterative":[16],"procedure":[17,34,61],"to":[18,53,63,102,128],"diagrams":[21],"generated":[22],"by":[23,94],"a":[24,64,97,103,126],"set":[25,99,105],"randomly":[28],"placed":[29],"on":[30],"plane.":[32],"The":[33,47,149,181,205,220],"implied":[35],"dividing":[36,48],"edges":[38],"cells":[41],"into":[42,112],"equal":[43],"or":[44],"parts.":[46],"were":[50],"then":[51],"used":[52],"construct":[54],"following":[56],"diagram.":[58],"Repeating":[59],"this":[60],"led":[62],"surprising":[65],"effect":[66],"positional":[69],"ordering":[70],"cells,":[73],"reminiscent":[74],"formation":[77],"lamellae":[79],"and":[80,86,152],"spherulites":[81],"in":[82,175,216],"linear":[83],"semi-crystalline":[84],"polymers":[85],"metallic":[87],"glasses.":[88],"Thus,":[89],"we":[90,108],"can":[91,109],"conclude":[92],"that":[93,131],"applying":[95],"even":[96],"simple":[98],"rules":[101],"seeds,":[107],"introduce":[110],"order":[111,215],"initially":[114],"disordered":[115],"system.":[116],"At":[117],"same":[119,198],"time,":[120],"Shannon":[122,140,150,182,206,231],"(Voronoi)":[123,141,207],"entropy":[124,142,151,183,208],"showed":[125],"tendency":[127],"attain":[129],"values":[130,171,228],"typical":[133],"for":[134],"completely":[135],"patterns;":[137],"thus,":[138],"does":[143],"not":[144,210],"distinguish":[145],"short-range":[147],"ordering.":[148],"continuous":[154,192],"measure":[155,193,213],"symmetry":[157,195],"patterns":[160,199,223],"demonstrated":[161,200],"distinct":[163],"asymptotic":[164,203],"behavior,":[165],"while":[166],"approaching":[167],"close":[169],"saturation":[170],"with":[172,185],"increase":[174],"number":[177,187],"iteration":[179],"steps.":[180],"grew":[184],"iterations,":[189],"whereas":[190],"opposite":[202],"behavior.":[204],"is":[209],"unambiguous":[212],"patterns.":[219],"more":[221],"symmetrical":[222],"may":[224],"demonstrate":[225],"higher":[227],"entropy.":[232]},"counts_by_year":[{"year":2026,"cited_by_count":4},{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":1}],"updated_date":"2026-07-15T18:14:33.161393","created_date":"2025-10-10T00:00:00"}