{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:KCGLR43LHA5IMAAJZCGLVZFLRL","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"60c214c9524dd610e99ebce145c0e3b02b1e9ca5427a9dbda98df6ed2dc1492e","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-03-24T11:41:50Z","title_canon_sha256":"28bd240ac9a65b323a4537b4e277a40e8e0fde389b15cfe9b8b55731b150e002"},"schema_version":"1.0","source":{"id":"1103.4738","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1103.4738","created_at":"2026-05-18T04:26:00Z"},{"alias_kind":"arxiv_version","alias_value":"1103.4738v1","created_at":"2026-05-18T04:26:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1103.4738","created_at":"2026-05-18T04:26:00Z"},{"alias_kind":"pith_short_12","alias_value":"KCGLR43LHA5I","created_at":"2026-05-18T12:26:32Z"},{"alias_kind":"pith_short_16","alias_value":"KCGLR43LHA5IMAAJ","created_at":"2026-05-18T12:26:32Z"},{"alias_kind":"pith_short_8","alias_value":"KCGLR43L","created_at":"2026-05-18T12:26:32Z"}],"graph_snapshots":[{"event_id":"sha256:a4e9dc6d18ff8d85730870bf0ee65b642aabf8b4df185b103f9287d11c9364eb","target":"graph","created_at":"2026-05-18T04:26:00Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Consider the homogeneous Boltzmann equation for Maxwellian molecules. We provide a new representation for its solution in the form of expectation of a random probability measure M. We also prove that the Fourier transform of M is a conditional characteristic function of a sum of independent random variables, given a suitable sigma-algebra. These facts are then used to prove a CLT for Maxwellian molecules, that is the statement of a necessary and sufficient condition for the weak convergence of the solution of the equation. Such a condition reduces to the finiteness of the second moment of the ","authors_text":"Emanuele Dolera, Eugenio Regazzini","cross_cats":["math-ph","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-03-24T11:41:50Z","title":"Probabilistic representation for the solution of the homogeneous Boltzmann equation for Maxwellian molecules"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1103.4738","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:6e7b8c0426b051e5b44e8c107bc7845e59d60cd105f2a106822119e718242d34","target":"record","created_at":"2026-05-18T04:26:00Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"60c214c9524dd610e99ebce145c0e3b02b1e9ca5427a9dbda98df6ed2dc1492e","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-03-24T11:41:50Z","title_canon_sha256":"28bd240ac9a65b323a4537b4e277a40e8e0fde389b15cfe9b8b55731b150e002"},"schema_version":"1.0","source":{"id":"1103.4738","kind":"arxiv","version":1}},"canonical_sha256":"508cb8f36b383a860009c88cbae4ab8ad42759e81882f431cbefa621491a240f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"508cb8f36b383a860009c88cbae4ab8ad42759e81882f431cbefa621491a240f","first_computed_at":"2026-05-18T04:26:00.952516Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:26:00.952516Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"X3OszeC+zQ0ilmX2kM8fBv4y7Dujur7Au/u39/KINddZ/96c3hfC5S4ZW7XQTnjqnONtCnKOit/j86d8aIu4Dw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:26:00.953003Z","signed_message":"canonical_sha256_bytes"},"source_id":"1103.4738","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6e7b8c0426b051e5b44e8c107bc7845e59d60cd105f2a106822119e718242d34","sha256:a4e9dc6d18ff8d85730870bf0ee65b642aabf8b4df185b103f9287d11c9364eb"],"state_sha256":"521f74d5a973c9926684c0530096dbaceaa68b377a0b110e207040d0a6fffe45"}