{"id":"https://openalex.org/W4221167381","doi":"https://doi.org/10.3390/e24050618","title":"Vibrational Entropy of Crystalline Solids from Covariance of Atomic Displacements","display_name":"Vibrational Entropy of Crystalline Solids from Covariance of Atomic Displacements","publication_year":2022,"publication_date":"2022-04-28","ids":{"openalex":"https://openalex.org/W4221167381","doi":"https://doi.org/10.3390/e24050618","pmid":"https://pubmed.ncbi.nlm.nih.gov/35626503"},"language":"en","primary_location":{"id":"doi:10.3390/e24050618","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e24050618","pdf_url":"https://www.mdpi.com/1099-4300/24/5/618/pdf?version=1651661605","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","pubmed"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/1099-4300/24/5/618/pdf?version=1651661605","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5101666256","display_name":"Yang Huang","orcid":"https://orcid.org/0000-0001-5255-0627"},"institutions":[{"id":"https://openalex.org/I74973139","display_name":"Carnegie Mellon University","ror":"https://ror.org/05x2bcf33","country_code":"US","type":"education","lineage":["https://openalex.org/I74973139"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Yang Huang","raw_affiliation_strings":["Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213, USA","institution_ids":["https://openalex.org/I74973139"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5019536088","display_name":"Michael Widom","orcid":"https://orcid.org/0000-0001-5972-5696"},"institutions":[{"id":"https://openalex.org/I74973139","display_name":"Carnegie Mellon University","ror":"https://ror.org/05x2bcf33","country_code":"US","type":"education","lineage":["https://openalex.org/I74973139"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Michael Widom","raw_affiliation_strings":["Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213, USA"],"raw_orcid":"https://orcid.org/0000-0001-5972-5696","affiliations":[{"raw_affiliation_string":"Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213, USA","institution_ids":["https://openalex.org/I74973139"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5019536088"],"corresponding_institution_ids":["https://openalex.org/I74973139"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2227},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2227},"fwci":2.0904,"has_fulltext":true,"cited_by_count":15,"citation_normalized_percentile":{"value":0.8552871,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"24","issue":"5","first_page":"618","last_page":"618"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10413","display_name":"High-pressure geophysics and materials","score":0.9995999932289124,"subfield":{"id":"https://openalex.org/subfields/1908","display_name":"Geophysics"},"field":{"id":"https://openalex.org/fields/19","display_name":"Earth and Planetary Sciences"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10413","display_name":"High-pressure geophysics and materials","score":0.9995999932289124,"subfield":{"id":"https://openalex.org/subfields/1908","display_name":"Geophysics"},"field":{"id":"https://openalex.org/fields/19","display_name":"Earth and Planetary Sciences"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12613","display_name":"X-ray Diffraction in Crystallography","score":0.9988999962806702,"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/T13645","display_name":"Thermodynamic and Structural Properties of Metals and Alloys","score":0.9965000152587891,"subfield":{"id":"https://openalex.org/subfields/2210","display_name":"Mechanical Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/covariance","display_name":"Covariance","score":0.7035192251205444},{"id":"https://openalex.org/keywords/statistical-physics","display_name":"Statistical physics","score":0.6434481143951416},{"id":"https://openalex.org/keywords/entropy","display_name":"Entropy (arrow of time)","score":0.5214440226554871},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.395347535610199},{"id":"https://openalex.org/keywords/thermodynamics","display_name":"Thermodynamics","score":0.3488466143608093},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.32706332206726074},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.2455308437347412}],"concepts":[{"id":"https://openalex.org/C178650346","wikidata":"https://www.wikidata.org/wiki/Q201984","display_name":"Covariance","level":2,"score":0.7035192251205444},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.6434481143951416},{"id":"https://openalex.org/C106301342","wikidata":"https://www.wikidata.org/wiki/Q4117933","display_name":"Entropy (arrow of time)","level":2,"score":0.5214440226554871},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.395347535610199},{"id":"https://openalex.org/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","level":1,"score":0.3488466143608093},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.32706332206726074},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.2455308437347412}],"mesh":[],"locations_count":8,"locations":[{"id":"doi:10.3390/e24050618","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e24050618","pdf_url":"https://www.mdpi.com/1099-4300/24/5/618/pdf?version=1651661605","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:35626503","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/35626503","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:arXiv.org:2203.13869","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2203.13869","pdf_url":"https://arxiv.org/pdf/2203.13869","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:pubmedcentral.nih.gov:9141984","is_oa":true,"landing_page_url":"https://www.ncbi.nlm.nih.gov/pmc/articles/9141984","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"},{"id":"pmh:oai:doaj.org/article:406a37a5c2064e528bfc64be43fda9a4","is_oa":true,"landing_page_url":"https://doaj.org/article/406a37a5c2064e528bfc64be43fda9a4","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 5, p 618 (2022)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/1099-4300/24/5/618/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/e24050618","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 24; Issue 5; Pages: 618","raw_type":"Text"},{"id":"pmh:oai:osti.gov:1865698","is_oa":true,"landing_page_url":"https://www.osti.gov/biblio/1865698","pdf_url":null,"source":{"id":"https://openalex.org/S4306402487","display_name":"OSTI OAI (U.S. Department of Energy Office of Scientific and Technical Information)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I139351228","host_organization_name":"Office of Scientific and Technical Information","host_organization_lineage":["https://openalex.org/I139351228"],"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":null},{"id":"pmh:oai:osti.gov:1865900","is_oa":true,"landing_page_url":"https://www.osti.gov/biblio/1865900","pdf_url":null,"source":{"id":"https://openalex.org/S4306402487","display_name":"OSTI OAI (U.S. Department of Energy Office of Scientific and Technical Information)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I139351228","host_organization_name":"Office of Scientific and Technical Information","host_organization_lineage":["https://openalex.org/I139351228"],"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":null}],"best_oa_location":{"id":"doi:10.3390/e24050618","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e24050618","pdf_url":"https://www.mdpi.com/1099-4300/24/5/618/pdf?version=1651661605","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":[{"id":"https://openalex.org/G122673668","display_name":null,"funder_award_id":"DE\u2013AC02\u201305CH11231","funder_id":"https://openalex.org/F4320317220","funder_display_name":"National Energy Research Scientific Computing Center"},{"id":"https://openalex.org/G1286236842","display_name":null,"funder_award_id":"-AC02-05CH11231","funder_id":"https://openalex.org/F4320306084","funder_display_name":"U.S. Department of Energy"},{"id":"https://openalex.org/G1677143136","display_name":null,"funder_award_id":"05CH11231","funder_id":"https://openalex.org/F4320317220","funder_display_name":"National Energy Research Scientific Computing Center"},{"id":"https://openalex.org/G3083819904","display_name":null,"funder_award_id":"05CH11231","funder_id":"https://openalex.org/F4320306084","funder_display_name":"U.S. Department of Energy"},{"id":"https://openalex.org/G419881185","display_name":null,"funder_award_id":"DEAC02\u201305CH11231","funder_id":"https://openalex.org/F4320332359","funder_display_name":"Office of Science"},{"id":"https://openalex.org/G42916606","display_name":null,"funder_award_id":"DE AC02 05CH11231","funder_id":"https://openalex.org/F4320306084","funder_display_name":"U.S. Department of Energy"},{"id":"https://openalex.org/G4501827968","display_name":null,"funder_award_id":"AC02-05CH11231","funder_id":"https://openalex.org/F4320332359","funder_display_name":"Office of Science"},{"id":"https://openalex.org/G498139845","display_name":null,"funder_award_id":"DE-AC02","funder_id":"https://openalex.org/F4320332359","funder_display_name":"Office of Science"},{"id":"https://openalex.org/G5076365615","display_name":null,"funder_award_id":"AC02-05CH11231","funder_id":"https://openalex.org/F4320317220","funder_display_name":"National Energy Research Scientific Computing Center"},{"id":"https://openalex.org/G5853243863","display_name":"THERMODYNAMIC STABILITY AND ELASTICITY OF HIGH ENTROPY ALLOYS","funder_award_id":"DESC0014506","funder_id":"https://openalex.org/F4320306084","funder_display_name":"U.S. Department of Energy"},{"id":"https://openalex.org/G6652958823","display_name":null,"funder_award_id":"05CH11231","funder_id":"https://openalex.org/F4320332359","funder_display_name":"Office of Science"},{"id":"https://openalex.org/G8436280960","display_name":null,"funder_award_id":"DE-SC0014506","funder_id":"https://openalex.org/F4320317220","funder_display_name":"National Energy Research Scientific Computing Center"},{"id":"https://openalex.org/G969889393","display_name":null,"funder_award_id":"DE-AC02-","funder_id":"https://openalex.org/F4320306084","funder_display_name":"U.S. Department of Energy"}],"funders":[{"id":"https://openalex.org/F4320306084","display_name":"U.S. Department of Energy","ror":"https://ror.org/01bj3aw27"},{"id":"https://openalex.org/F4320317220","display_name":"National Energy Research Scientific Computing Center","ror":"https://ror.org/05v3mvq14"},{"id":"https://openalex.org/F4320332359","display_name":"Office of Science","ror":"https://ror.org/00mmn6b08"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4221167381.pdf","grobid_xml":"https://content.openalex.org/works/W4221167381.grobid-xml"},"referenced_works_count":53,"referenced_works":["https://openalex.org/W770006305","https://openalex.org/W832976576","https://openalex.org/W1167040955","https://openalex.org/W1480344782","https://openalex.org/W1538796956","https://openalex.org/W1965004209","https://openalex.org/W1965127276","https://openalex.org/W1967091917","https://openalex.org/W1973938464","https://openalex.org/W1978191758","https://openalex.org/W1979544533","https://openalex.org/W1981368803","https://openalex.org/W1989272457","https://openalex.org/W1993300454","https://openalex.org/W1995875735","https://openalex.org/W1999463389","https://openalex.org/W2006214307","https://openalex.org/W2024400312","https://openalex.org/W2038601818","https://openalex.org/W2051923595","https://openalex.org/W2052637367","https://openalex.org/W2058241222","https://openalex.org/W2069988019","https://openalex.org/W2071924280","https://openalex.org/W2073945656","https://openalex.org/W2078492700","https://openalex.org/W2083222334","https://openalex.org/W2083372231","https://openalex.org/W2097650833","https://openalex.org/W2126706680","https://openalex.org/W2137605538","https://openalex.org/W2527902837","https://openalex.org/W2528319605","https://openalex.org/W2583890051","https://openalex.org/W2586282649","https://openalex.org/W2745356055","https://openalex.org/W2761796810","https://openalex.org/W2884500776","https://openalex.org/W2890628255","https://openalex.org/W2902136118","https://openalex.org/W2914613295","https://openalex.org/W2914881579","https://openalex.org/W2940010487","https://openalex.org/W2951697263","https://openalex.org/W2963657244","https://openalex.org/W2963940834","https://openalex.org/W2964327629","https://openalex.org/W3100203070","https://openalex.org/W3108237574","https://openalex.org/W3131202720","https://openalex.org/W3159976237","https://openalex.org/W4224987962","https://openalex.org/W4249855967"],"related_works":["https://openalex.org/W4391375266","https://openalex.org/W1979597421","https://openalex.org/W2007980826","https://openalex.org/W2061531152","https://openalex.org/W3002753104","https://openalex.org/W2077600819","https://openalex.org/W2142036596","https://openalex.org/W2072657027","https://openalex.org/W2600246793","https://openalex.org/W2180033908"],"abstract_inverted_index":{"The":[0],"vibrational":[1,133],"entropy":[2,46,147,173],"of":[3,14,25,38,51,57,88,96,122,135,139],"a":[4,54,171],"solid":[5],"at":[6,28],"finite":[7,29],"temperature":[8,30],"is":[9],"investigated":[10],"from":[11,31,47,137],"the":[12,35,44,48,81,85,92,119,123,126,140,146,182,191],"perspective":[13],"information":[15],"theory.":[16],"Ab":[17],"initio":[18],"molecular":[19],"dynamics":[20],"(AIMD)":[21],"simulations":[22],"generate":[23],"ensembles":[24],"atomic":[26,39,61],"configurations":[27],"which":[32],"we":[33,90,129,168,187],"obtain":[34,76],"N-body":[36],"distribution":[37],"displacements,":[40],"\u03c1N.":[41],"We":[42,99],"calculate":[43],"information-theoretic":[45],"expectation":[49],"value":[50],"ln\u03c1N.":[52],"At":[53,84],"first":[55],"level":[56,87],"approximation,":[58],"treating":[59],"individual":[60],"displacements":[62],"independently,":[63],"our":[64,176],"method":[65,161,177],"may":[66],"be":[67],"applied":[68],"using":[69],"Debye\u2013Waller":[70],"B-factors,":[71],"allowing":[72],"diffraction":[73],"experiments":[74],"to":[75,103,157,164],"an":[77,131],"upper":[78],"bound":[79],"on":[80],"thermodynamic":[82],"entropy.":[83],"next":[86],"approximation":[89,184],"correct":[91],"overestimation":[93],"through":[94],"inclusion":[95],"displacement":[97],"covariances.":[98],"apply":[100],"this":[101],"approach":[102],"elemental":[104],"body-centered":[105],"cubic":[106,110],"sodium":[107],"and":[108,143],"face-centered":[109],"aluminum,":[111],"showing":[112],"good":[113,152],"agreement":[114,153],"with":[115,154],"experimental":[116],"values":[117],"above":[118],"Debye":[120,127],"temperatures":[121],"metals.":[124],"Below":[125],"temperatures,":[128],"extract":[130],"effective":[132],"density":[134],"states":[136],"eigenvalues":[138],"covariance":[141],"matrix,":[142],"then":[144],"evaluate":[145],"quantum":[148],"mechanically,":[149],"again":[150],"yielding":[151],"experiment":[155],"down":[156],"low":[158],"temperatures.":[159],"Our":[160],"readily":[162],"generalizes":[163],"complex":[165],"solids,":[166],"as":[167,186],"demonstrate":[169,188],"for":[170],"high":[172],"alloy.":[174],"Further,":[175],"applies":[178],"in":[179,194],"cases":[180],"where":[181],"quasiharmonic":[183],"fails,":[185],"by":[189],"calculating":[190],"HCP/BCC":[192],"transition":[193],"Ti.":[195]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":6},{"year":2024,"cited_by_count":5},{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":1}],"updated_date":"2026-08-11T07:18:39.950985","created_date":"2025-10-10T00:00:00"}
