{"id":"https://openalex.org/W7214508239","doi":"https://doi.org/10.48550/arxiv.2609.29566","title":"A Periodic Long-Time Boltzmann--Grad Limit in Every Fixed Dimension $d\\ge 4$","display_name":"A Periodic Long-Time Boltzmann--Grad Limit in Every Fixed Dimension $d\\ge 4$","publication_year":2026,"publication_date":"2026-08-26","ids":{"openalex":"https://openalex.org/W7214508239","doi":"https://doi.org/10.48550/arxiv.2609.29566"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.29566","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.29566","pdf_url":"https://arxiv.org/pdf/2609.29566","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,"listed_in":[],"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"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2609.29566","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5147446154","display_name":"Zhixu Hua","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Hua, Zhixu","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5147446154"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11012","display_name":"Gas Dynamics and Kinetic Theory","score":0.7505000233650208,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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/T11012","display_name":"Gas Dynamics and Kinetic Theory","score":0.7505000233650208,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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/T11152","display_name":"Stochastic processes and statistical mechanics","score":0.18029999732971191,"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"}},{"id":"https://openalex.org/T10588","display_name":"Mathematical Dynamics and Fractals","score":0.011300000362098217,"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/boltzmann-grad-limit","display_name":"Boltzmann-Grad limit","score":0.9459999799728394},{"id":"https://openalex.org/keywords/hard-spheres","display_name":"hard spheres","score":0.9039999842643738},{"id":"https://openalex.org/keywords/periodic-boundary-conditions","display_name":"periodic boundary conditions","score":0.7059999704360962},{"id":"https://openalex.org/keywords/long-time-behavior","display_name":"long-time behavior","score":0.8320000171661377},{"id":"https://openalex.org/keywords/many-body-correlations","display_name":"many-body correlations","score":0.5360000133514404},{"id":"https://openalex.org/keywords/higher-spatial-dimensions","display_name":"higher spatial dimensions","score":0.5490000247955322}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7763000130653381},{"id":"https://openalex.org/C45340560","wikidata":"https://www.wikidata.org/wiki/Q215382","display_name":"Disjoint sets","level":2,"score":0.6322000026702881},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.4713999927043915},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.46869999170303345},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.46389999985694885},{"id":"https://openalex.org/C151201525","wikidata":"https://www.wikidata.org/wiki/Q177239","display_name":"Limit (mathematics)","level":2,"score":0.46380001306533813},{"id":"https://openalex.org/C129782007","wikidata":"https://www.wikidata.org/wiki/Q162886","display_name":"Euclidean geometry","level":2,"score":0.4189999997615814},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.41670000553131104},{"id":"https://openalex.org/C136886441","wikidata":"https://www.wikidata.org/wiki/Q926129","display_name":"Normalization (sociology)","level":2,"score":0.3982999920845032},{"id":"https://openalex.org/C186450821","wikidata":"https://www.wikidata.org/wiki/Q17295","display_name":"Euclidean space","level":2,"score":0.39100000262260437},{"id":"https://openalex.org/C158379750","wikidata":"https://www.wikidata.org/wiki/Q214111","display_name":"Network packet","level":2,"score":0.3896999955177307},{"id":"https://openalex.org/C121704057","wikidata":"https://www.wikidata.org/wiki/Q352070","display_name":"Collision","level":2,"score":0.3720000088214874},{"id":"https://openalex.org/C120174047","wikidata":"https://www.wikidata.org/wiki/Q847073","display_name":"Euclidean distance","level":2,"score":0.3450999855995178},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.3449000120162964},{"id":"https://openalex.org/C184898388","wikidata":"https://www.wikidata.org/wiki/Q1435712","display_name":"Pairwise comparison","level":2,"score":0.33399999141693115},{"id":"https://openalex.org/C58315980","wikidata":"https://www.wikidata.org/wiki/Q179436","display_name":"Asymptote","level":2,"score":0.3255999982357025},{"id":"https://openalex.org/C194147245","wikidata":"https://www.wikidata.org/wiki/Q1076368","display_name":"Chord (peer-to-peer)","level":2,"score":0.32100000977516174},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.30720001459121704},{"id":"https://openalex.org/C108598597","wikidata":"https://www.wikidata.org/wiki/Q124255","display_name":"Conic section","level":2,"score":0.3043999969959259},{"id":"https://openalex.org/C72422203","wikidata":"https://www.wikidata.org/wiki/Q7392545","display_name":"SPHERES","level":2,"score":0.2980000078678131},{"id":"https://openalex.org/C150397156","wikidata":"https://www.wikidata.org/wiki/Q591310","display_name":"General position","level":2,"score":0.29760000109672546},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.29440000653266907},{"id":"https://openalex.org/C182124507","wikidata":"https://www.wikidata.org/wiki/Q166154","display_name":"Line segment","level":2,"score":0.2928999960422516},{"id":"https://openalex.org/C182964748","wikidata":"https://www.wikidata.org/wiki/Q208216","display_name":"Triangle inequality","level":2,"score":0.29269999265670776},{"id":"https://openalex.org/C145691206","wikidata":"https://www.wikidata.org/wiki/Q747980","display_name":"Polytope","level":2,"score":0.28870001435279846},{"id":"https://openalex.org/C13355873","wikidata":"https://www.wikidata.org/wiki/Q2920850","display_name":"Connection (principal bundle)","level":2,"score":0.2840000092983246},{"id":"https://openalex.org/C75432250","wikidata":"https://www.wikidata.org/wiki/Q29654355","display_name":"Real line","level":2,"score":0.26409998536109924},{"id":"https://openalex.org/C191795146","wikidata":"https://www.wikidata.org/wiki/Q3878446","display_name":"Norm (philosophy)","level":2,"score":0.25949999690055847}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.29566","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.29566","pdf_url":"https://arxiv.org/pdf/2609.29566","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,"listed_in":[],"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":"doi:10.48550/arxiv.2609.29566","doi":"https://doi.org/10.48550/arxiv.2609.29566","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.29566","pdf_url":null,"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,"listed_in":[],"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":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.29566","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.29566","pdf_url":"https://arxiv.org/pdf/2609.29566","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,"listed_in":[],"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"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7214508239.pdf","grobid_xml":"https://content.openalex.org/works/W7214508239.grobid-xml"},"referenced_works_count":2,"referenced_works":["https://openalex.org/W4406022324","https://openalex.org/W4415085886"],"related_works":["https://openalex.org/W7204257543","https://openalex.org/W2025608992","https://openalex.org/W3105715855","https://openalex.org/W2731830792","https://openalex.org/W1966335976","https://openalex.org/W1612303707","https://openalex.org/W2130650797","https://openalex.org/W1842640703","https://openalex.org/W1976492544","https://openalex.org/W2090271561"],"abstract_inverted_index":{"We":[0,76],"prove":[1],"a":[2,80,101,115,123,204,224],"periodic":[3,150],"long-time":[4],"Boltzmann--Grad":[5],"limit":[6],"for":[7,40,83,141,167,177],"hard":[8],"spheres":[9],"in":[10,28,66,73],"every":[11,178,195],"fixed":[12,179],"spatial":[13],"dimension":[14],"$d\\ge4$.":[15],"Under":[16],"the":[17,20,23,31,51,89,106,109,139,154,159,173,183,188,209,215],"assumptions":[18],"of":[19],"main":[21],"theorem,":[22],"rescaled":[24],"$s$-particle":[25],"correlations":[26],"converge":[27],"$L^1$":[29],"to":[30],"corresponding":[32],"tensor-product":[33],"Boltzmann":[34],"profile":[35],"at":[36,198],"rate":[37],"$\\varepsilon^{1/(400d)}$,":[38],"uniformly":[39],"$1\\le":[41],"s\\le|\\log\\varepsilon|$":[42],"and":[43,53,69,153,208],"$0\\le":[44],"t\\le":[45],"t_{\\rm":[46],"fin}$.":[47],"In":[48],"particular,":[49],"when":[50],"activity":[52],"weighted":[54],"solution":[55],"bounds":[56],"are":[57],"fixed,":[58],"$t_{\\rm":[59],"fin}=O(\\log|\\log\\varepsilon|)$.":[60],"The":[61],"componentwise":[62],"long-bond":[63],"estimate":[64,82,140,222],"used":[65],"dimensions":[67],"two":[68,84,91],"three":[70],"is":[71,214],"insufficient":[72],"higher":[74],"dimension.":[75],"replace":[77],"it":[78],"by":[79],"joint":[81,144,170],"connected":[85,190],"time":[86,185],"sublayers,":[87],"selecting":[88],"first":[90,216],"lower":[92],"collisions":[93],"as":[94],"landing":[95,98],"roots.":[96],"Disjoint":[97],"edges":[99,107],"yield":[100],"direct":[102],"tangential":[103],"frame.":[104],"When":[105],"overlap,":[108],"newly":[110],"appearing":[111],"particle":[112],"line":[113],"contains":[114],"collision-free":[116],"chord":[117],"whose":[118],"radial":[119],"relative":[120],"speed":[121],"produces":[122],"rank-$(d-1)$":[124],"positive":[125,129],"Schur":[126],"factor.":[127],"A":[128,146,163,219],"Jacobi-network":[130],"argument":[131],"prevents":[132],"focusing,":[133],"while":[134],"bounded-degree":[135],"finite-fibre":[136],"coarea":[137],"globalizes":[138],"arbitrary":[142],"nonnegative":[143],"kernels.":[145],"first-failure":[147],"construction":[148],"repairs":[149],"double":[151],"overlaps,":[152],"resulting":[155],"packet":[156,193,196],"bound":[157],"closes":[158],"exceptional":[160],"top-layer":[161],"contribution.":[162],"paid":[164],"epoch":[165],"restart":[166],"sealed":[168],"complete":[169],"kernels,":[171],"using":[172],"fixed-word":[174],"multi-landing":[175],"operator":[176,210],"integer":[180],"$k\\ge1$,":[181],"yields":[182],"stated":[184],"range.":[186],"Within":[187],"regular":[189],"full":[191],"two-sublayer":[192],"class,":[194],"with":[197],"least":[199],"$2k-1$":[200],"physical":[201],"lines":[202],"admits":[203],"canonical":[205],"$k$-birth":[206],"flag":[207],"factor":[211],"$\\varepsilon^{(d-1)(k-1)}\\varepsilon_*^{-k(d-1)}$;":[212],"$k=2$":[213],"gain-producing":[217],"case.":[218],"complementary":[220],"full-chord":[221],"provides":[223],"coarse":[225],"fallback.":[226]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-27T00:00:00"}