{"id":"https://openalex.org/W4405538190","doi":"https://doi.org/10.3390/e26121106","title":"The Boltzmann Equation and Its Place in the Edifice of Statistical Mechanics","display_name":"The Boltzmann Equation and Its Place in the Edifice of Statistical Mechanics","publication_year":2024,"publication_date":"2024-12-18","ids":{"openalex":"https://openalex.org/W4405538190","doi":"https://doi.org/10.3390/e26121106","pmid":"https://pubmed.ncbi.nlm.nih.gov/39766735"},"language":"en","primary_location":{"id":"doi:10.3390/e26121106","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e26121106","pdf_url":"https://www.mdpi.com/1099-4300/26/12/1106/pdf?version=1734526705","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/1099-4300/26/12/1106/pdf?version=1734526705","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5109247130","display_name":"Charlotte Werndl","orcid":"https://orcid.org/0000-0003-2980-8436"},"institutions":[{"id":"https://openalex.org/I182212641","display_name":"University of Salzburg","ror":"https://ror.org/05gs8cd61","country_code":"AT","type":"education","lineage":["https://openalex.org/I182212641"]}],"countries":["AT"],"is_corresponding":false,"raw_author_name":"Charlotte Werndl","raw_affiliation_strings":["Department of Philosophy, Faculty of Social Sciences, University of Salzburg, Franziskanergasse 1, 5020 Salzburg, Austria"],"raw_orcid":"https://orcid.org/0000-0003-2980-8436","affiliations":[{"raw_affiliation_string":"Department of Philosophy, Faculty of Social Sciences, University of Salzburg, Franziskanergasse 1, 5020 Salzburg, Austria","institution_ids":["https://openalex.org/I182212641"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5077859671","display_name":"Roman Frigg","orcid":"https://orcid.org/0000-0003-0812-0907"},"institutions":[{"id":"https://openalex.org/I909854389","display_name":"London School of Economics and Political Science","ror":"https://ror.org/0090zs177","country_code":"GB","type":"education","lineage":["https://openalex.org/I124357947","https://openalex.org/I909854389"]}],"countries":["GB"],"is_corresponding":true,"raw_author_name":"Roman Frigg","raw_affiliation_strings":["Department of Philosophy, Logic and Scientific Method, London School of Economics, Houghton Street, London WC2A 2AE, UK"],"raw_orcid":"https://orcid.org/0000-0003-0812-0907","affiliations":[{"raw_affiliation_string":"Department of Philosophy, Logic and Scientific Method, London School of Economics, Houghton Street, London WC2A 2AE, UK","institution_ids":["https://openalex.org/I909854389"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5077859671"],"corresponding_institution_ids":["https://openalex.org/I909854389"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":0.8591,"has_fulltext":true,"cited_by_count":3,"citation_normalized_percentile":{"value":0.73495009,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":91,"max":99},"biblio":{"volume":"26","issue":"12","first_page":"1106","last_page":"1106"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10591","display_name":"Theoretical and Computational Physics","score":0.9990000128746033,"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"}},"topics":[{"id":"https://openalex.org/T10591","display_name":"Theoretical and Computational Physics","score":0.9990000128746033,"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/T11520","display_name":"Advanced Thermodynamics and Statistical Mechanics","score":0.9987999796867371,"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/T12261","display_name":"Statistical Mechanics and Entropy","score":0.9959999918937683,"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/bbgky-hierarchy","display_name":"BBGKY hierarchy","score":0.8660099506378174},{"id":"https://openalex.org/keywords/statistical-mechanics","display_name":"Statistical mechanics","score":0.7059895396232605},{"id":"https://openalex.org/keywords/hierarchy","display_name":"Hierarchy","score":0.6411628127098083},{"id":"https://openalex.org/keywords/statistical-physics","display_name":"Statistical physics","score":0.5546678900718689},{"id":"https://openalex.org/keywords/boltzmann-constant","display_name":"Boltzmann constant","score":0.5074431300163269},{"id":"https://openalex.org/keywords/boltzmann-equation","display_name":"Boltzmann equation","score":0.4357941150665283},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.424850732088089},{"id":"https://openalex.org/keywords/boltzmanns-entropy-formula","display_name":"Boltzmann's entropy formula","score":0.4162234663963318},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.29959607124328613},{"id":"https://openalex.org/keywords/distribution-function","display_name":"Distribution function","score":0.2139500081539154},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.12720003724098206}],"concepts":[{"id":"https://openalex.org/C167312068","wikidata":"https://www.wikidata.org/wiki/Q653679","display_name":"BBGKY hierarchy","level":3,"score":0.8660099506378174},{"id":"https://openalex.org/C99874945","wikidata":"https://www.wikidata.org/wiki/Q188715","display_name":"Statistical mechanics","level":2,"score":0.7059895396232605},{"id":"https://openalex.org/C31170391","wikidata":"https://www.wikidata.org/wiki/Q188619","display_name":"Hierarchy","level":2,"score":0.6411628127098083},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.5546678900718689},{"id":"https://openalex.org/C35304006","wikidata":"https://www.wikidata.org/wiki/Q5962","display_name":"Boltzmann constant","level":2,"score":0.5074431300163269},{"id":"https://openalex.org/C165995430","wikidata":"https://www.wikidata.org/wiki/Q891653","display_name":"Boltzmann equation","level":2,"score":0.4357941150665283},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.424850732088089},{"id":"https://openalex.org/C33932047","wikidata":"https://www.wikidata.org/wiki/Q375553","display_name":"Boltzmann's entropy formula","level":3,"score":0.4162234663963318},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.29959607124328613},{"id":"https://openalex.org/C186603090","wikidata":"https://www.wikidata.org/wiki/Q55505926","display_name":"Distribution function","level":2,"score":0.2139500081539154},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.12720003724098206},{"id":"https://openalex.org/C34447519","wikidata":"https://www.wikidata.org/wiki/Q179522","display_name":"Market economy","level":1,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0}],"mesh":[],"locations_count":6,"locations":[{"id":"doi:10.3390/e26121106","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e26121106","pdf_url":"https://www.mdpi.com/1099-4300/26/12/1106/pdf?version=1734526705","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:39766735","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/39766735","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:philsci-archive.pitt.edu:25679","is_oa":true,"landing_page_url":null,"pdf_url":"https://philsci-archive.pitt.edu/25679/1/entropy-26-01106.pdf","source":{"id":"https://openalex.org/S4306401282","display_name":"PhilSci-Archive (University of Pittsburgh)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I170201317","host_organization_name":"University of Pittsburgh","host_organization_lineage":["https://openalex.org/I170201317"],"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":"NonPeerReviewed"},{"id":"pmh:oai:eprints.lse.ac.uk:126400","is_oa":true,"landing_page_url":"https://orcid.org/0000-0003-0812-0907>","pdf_url":"http://eprints.lse.ac.uk/126400/1/entropy-26-01106.pdf","source":{"id":"https://openalex.org/S4306401593","display_name":"London School of Economics and Political Science Research Online (London School of Economics and Political Science)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I909854389","host_organization_name":"London School of Economics and Political Science","host_organization_lineage":["https://openalex.org/I909854389"],"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":"PeerReviewed"},{"id":"pmh:oai:pubmedcentral.nih.gov:11675967","is_oa":true,"landing_page_url":"https://www.ncbi.nlm.nih.gov/pmc/articles/11675967","pdf_url":"https://pmc.ncbi.nlm.nih.gov/articles/PMC11675967/pdf/entropy-26-01106.pdf","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:513737e006b146139df30d5a7afb0df4","is_oa":false,"landing_page_url":"https://doaj.org/article/513737e006b146139df30d5a7afb0df4","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":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Entropy, Vol 26, Iss 12, p 1106 (2024)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.3390/e26121106","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e26121106","pdf_url":"https://www.mdpi.com/1099-4300/26/12/1106/pdf?version=1734526705","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":[{"id":"https://metadata.un.org/sdg/11","display_name":"Sustainable cities and communities","score":0.46000000834465027}],"awards":[{"id":"https://openalex.org/G6972899807","display_name":null,"funder_award_id":"10.55776/CO3","funder_id":"https://openalex.org/F4320321181","funder_display_name":"Austrian Science Fund"}],"funders":[{"id":"https://openalex.org/F4320321181","display_name":"Austrian Science Fund","ror":"https://ror.org/013tf3c58"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4405538190.pdf","grobid_xml":"https://content.openalex.org/works/W4405538190.grobid-xml"},"referenced_works_count":30,"referenced_works":["https://openalex.org/W81959524","https://openalex.org/W1518063347","https://openalex.org/W1561968718","https://openalex.org/W1568056205","https://openalex.org/W1570037176","https://openalex.org/W1583882625","https://openalex.org/W1932520202","https://openalex.org/W1954414299","https://openalex.org/W2038829389","https://openalex.org/W2047187412","https://openalex.org/W2048220528","https://openalex.org/W2059108151","https://openalex.org/W2086944521","https://openalex.org/W2113437367","https://openalex.org/W2120182139","https://openalex.org/W2179607442","https://openalex.org/W2354749622","https://openalex.org/W2482589551","https://openalex.org/W2496653833","https://openalex.org/W2789397391","https://openalex.org/W2808163742","https://openalex.org/W3099695579","https://openalex.org/W4206976429","https://openalex.org/W4230015191","https://openalex.org/W4238215780","https://openalex.org/W4243802917","https://openalex.org/W4245156102","https://openalex.org/W4386232904","https://openalex.org/W4389579947","https://openalex.org/W6776369805"],"related_works":["https://openalex.org/W1974055559","https://openalex.org/W3003415225","https://openalex.org/W4238106424","https://openalex.org/W3201567749","https://openalex.org/W2055744950","https://openalex.org/W593586691","https://openalex.org/W2790803144","https://openalex.org/W2009103811","https://openalex.org/W892764504","https://openalex.org/W2007308659"],"abstract_inverted_index":{"It":[0,21],"is":[1,72,106],"customary":[2],"to":[3,13,63],"classify":[4],"approaches":[5],"in":[6,56,109],"statistical":[7],"mechanics":[8],"(SM)":[9],"as":[10],"belonging":[11],"either":[12,32],"Boltzmanninan":[14],"SM":[15,19],"(BSM)":[16],"or":[17],"Gibbsian":[18],"(GSM).":[20],"is,":[22],"however,":[23],"unclear":[24],"how":[25],"the":[26,38,82,91,96,99,104,110,121,125],"Boltzmann":[27],"equation":[28],"(BE)":[29],"fits":[30],"into":[31],"of":[33,49,76,98,124],"these":[34],"approaches.":[35],"To":[36,80],"discuss":[37,81],"relation":[39,83],"between":[40,84],"BE":[41,71,85,100],"and":[42,66,86,94],"BSM,":[43],"we":[44,67,88],"first":[45],"present":[46],"a":[47,73],"version":[48,97],"BSM":[50,77],"that":[51,57,70,101,113],"differs":[52],"from":[53,103],"standard":[54],"presentation":[55],"it":[58,114],"uses":[59],"local":[60],"field":[61],"variables":[62],"individuate":[64],"macro-states,":[65],"then":[68],"show":[69],"special":[74],"case":[75],"thus":[78],"understood.":[79],"GSM,":[87],"focus":[89],"on":[90,120],"BBGKY":[92],"hierarchy":[93,105],"note":[95],"follows":[102],"\u201cGibbsian\u201d":[107],"only":[108],"minimal":[111],"sense":[112],"operates":[115],"with":[116],"an":[117],"invariant":[118],"measure":[119],"state":[122],"space":[123],"full":[126],"system.":[127]},"counts_by_year":[{"year":2026,"cited_by_count":2},{"year":2025,"cited_by_count":1}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}