{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2020:N3ASSRENESHCZMIJBEMZPCGIUI","short_pith_number":"pith:N3ASSREN","schema_version":"1.0","canonical_sha256":"6ec129448d248e2cb10909199788c8a23ebc90f7da440bcee5b60d3dd7ae909b","source":{"kind":"arxiv","id":"2001.01546","version":2},"attestation_state":"computed","paper":{"title":"The mean-field limit of quantum Bose gases at positive temperature","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.MP","math.PR"],"primary_cat":"math-ph","authors_text":"Antti Knowles, Benjamin Schlein, J\\\"urg Fr\\\"ohlich, Vedran Sohinger","submitted_at":"2020-01-06T13:17:58Z","abstract_excerpt":"We prove that the grand canonical Gibbs state of an interacting quantum Bose gas converges to the Gibbs measure of a nonlinear Schr\\\"odinger equation in the mean-field limit, where the density of the gas becomes large and the interaction strength is proportional to the inverse density. Our results hold in dimensions $d \\leq 3$. For $d > 1$ the Gibbs measure is supported on distributions of negative regularity and we have to renormalize the interaction. More precisely, we prove the convergence of the relative partition function and of the reduced density matrices in the $L^r$-norm with optimal "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2001.01546","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2020-01-06T13:17:58Z","cross_cats_sorted":["math.AP","math.MP","math.PR"],"title_canon_sha256":"816306add0c74c8508fd5c8b6350708f8c5641c9684ddc2b1867b15fc9dd5c3a","abstract_canon_sha256":"8231cd07f1ac074aeb1f4581bf171f0448165a7a6beec4ef04488c4464e3a21f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T02:50:38.062707Z","signature_b64":"J+S10c+krVpShEzQ1W0VNPzZ+wqE7XaaiTL9XGrqsNFrtgK1gj7XPiaGDZ1ik9+uisPeagM4tINuDpdcR1DuAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6ec129448d248e2cb10909199788c8a23ebc90f7da440bcee5b60d3dd7ae909b","last_reissued_at":"2026-07-05T02:50:38.062284Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T02:50:38.062284Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The mean-field limit of quantum Bose gases at positive temperature","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.MP","math.PR"],"primary_cat":"math-ph","authors_text":"Antti Knowles, Benjamin Schlein, J\\\"urg Fr\\\"ohlich, Vedran Sohinger","submitted_at":"2020-01-06T13:17:58Z","abstract_excerpt":"We prove that the grand canonical Gibbs state of an interacting quantum Bose gas converges to the Gibbs measure of a nonlinear Schr\\\"odinger equation in the mean-field limit, where the density of the gas becomes large and the interaction strength is proportional to the inverse density. Our results hold in dimensions $d \\leq 3$. For $d > 1$ the Gibbs measure is supported on distributions of negative regularity and we have to renormalize the interaction. More precisely, we prove the convergence of the relative partition function and of the reduced density matrices in the $L^r$-norm with optimal "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2001.01546","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2001.01546/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"2001.01546","created_at":"2026-07-05T02:50:38.062349+00:00"},{"alias_kind":"arxiv_version","alias_value":"2001.01546v2","created_at":"2026-07-05T02:50:38.062349+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2001.01546","created_at":"2026-07-05T02:50:38.062349+00:00"},{"alias_kind":"pith_short_12","alias_value":"N3ASSRENESHC","created_at":"2026-07-05T02:50:38.062349+00:00"},{"alias_kind":"pith_short_16","alias_value":"N3ASSRENESHCZMIJ","created_at":"2026-07-05T02:50:38.062349+00:00"},{"alias_kind":"pith_short_8","alias_value":"N3ASSREN","created_at":"2026-07-05T02:50:38.062349+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/N3ASSRENESHCZMIJBEMZPCGIUI","json":"https://pith.science/pith/N3ASSRENESHCZMIJBEMZPCGIUI.json","graph_json":"https://pith.science/api/pith-number/N3ASSRENESHCZMIJBEMZPCGIUI/graph.json","events_json":"https://pith.science/api/pith-number/N3ASSRENESHCZMIJBEMZPCGIUI/events.json","paper":"https://pith.science/paper/N3ASSREN"},"agent_actions":{"view_html":"https://pith.science/pith/N3ASSRENESHCZMIJBEMZPCGIUI","download_json":"https://pith.science/pith/N3ASSRENESHCZMIJBEMZPCGIUI.json","view_paper":"https://pith.science/paper/N3ASSREN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2001.01546&json=true","fetch_graph":"https://pith.science/api/pith-number/N3ASSRENESHCZMIJBEMZPCGIUI/graph.json","fetch_events":"https://pith.science/api/pith-number/N3ASSRENESHCZMIJBEMZPCGIUI/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/N3ASSRENESHCZMIJBEMZPCGIUI/action/timestamp_anchor","attest_storage":"https://pith.science/pith/N3ASSRENESHCZMIJBEMZPCGIUI/action/storage_attestation","attest_author":"https://pith.science/pith/N3ASSRENESHCZMIJBEMZPCGIUI/action/author_attestation","sign_citation":"https://pith.science/pith/N3ASSRENESHCZMIJBEMZPCGIUI/action/citation_signature","submit_replication":"https://pith.science/pith/N3ASSRENESHCZMIJBEMZPCGIUI/action/replication_record"}},"created_at":"2026-07-05T02:50:38.062349+00:00","updated_at":"2026-07-05T02:50:38.062349+00:00"}