{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:TDFNHFBKU7PPQGH5FED7HXT3CM","short_pith_number":"pith:TDFNHFBK","canonical_record":{"source":{"id":"1401.0829","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-01-04T17:19:14Z","cross_cats_sorted":["math-ph","math.AP","math.MP"],"title_canon_sha256":"1d3aa14fabfca478560478d70e2a83b4584bbced649d80207109b1bd9886460c","abstract_canon_sha256":"d0b29b8321bebd60eec168fbf7d214bb485b2552912a4ab2276137109aacd301"},"schema_version":"1.0"},"canonical_sha256":"98cad3942aa7def818fd2907f3de7b132d378639c4258981a72583ac4a628ba6","source":{"kind":"arxiv","id":"1401.0829","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1401.0829","created_at":"2026-05-17T23:54:07Z"},{"alias_kind":"arxiv_version","alias_value":"1401.0829v2","created_at":"2026-05-17T23:54:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1401.0829","created_at":"2026-05-17T23:54:07Z"},{"alias_kind":"pith_short_12","alias_value":"TDFNHFBKU7PP","created_at":"2026-05-18T12:28:49Z"},{"alias_kind":"pith_short_16","alias_value":"TDFNHFBKU7PPQGH5","created_at":"2026-05-18T12:28:49Z"},{"alias_kind":"pith_short_8","alias_value":"TDFNHFBK","created_at":"2026-05-18T12:28:49Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:TDFNHFBKU7PPQGH5FED7HXT3CM","target":"record","payload":{"canonical_record":{"source":{"id":"1401.0829","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-01-04T17:19:14Z","cross_cats_sorted":["math-ph","math.AP","math.MP"],"title_canon_sha256":"1d3aa14fabfca478560478d70e2a83b4584bbced649d80207109b1bd9886460c","abstract_canon_sha256":"d0b29b8321bebd60eec168fbf7d214bb485b2552912a4ab2276137109aacd301"},"schema_version":"1.0"},"canonical_sha256":"98cad3942aa7def818fd2907f3de7b132d378639c4258981a72583ac4a628ba6","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:54:07.919125Z","signature_b64":"TaqXSUFmtb0x0HmUfzV5fTQKReo8ecj54YMpPjzM9hQ3hg/MI0FNqRkpU+fZwo3tt1ux/gqDQeQpfCnLvTDLBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"98cad3942aa7def818fd2907f3de7b132d378639c4258981a72583ac4a628ba6","last_reissued_at":"2026-05-17T23:54:07.918691Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:54:07.918691Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1401.0829","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:54:07Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"N7YFgxDPKGt3CBCdjrNzwWbtBTN5nCeWVz7lf2Qyur+NodiqO5Ys9LoLwCx8sB09azApSVtkthSDKmeXuB74Dg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T07:23:24.174478Z"},"content_sha256":"ce3b31b9fb3f6aaade9464c883e51fb80b2922c635a26c45bc5f665f60dbe25c","schema_version":"1.0","event_id":"sha256:ce3b31b9fb3f6aaade9464c883e51fb80b2922c635a26c45bc5f665f60dbe25c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:TDFNHFBKU7PPQGH5FED7HXT3CM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Coagulation and diffusion: a probabilistic perspective on the Smoluchowski PDE","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.AP","math.MP"],"primary_cat":"math.PR","authors_text":"Alan Hammond","submitted_at":"2014-01-04T17:19:14Z","abstract_excerpt":"The Smoluchowski coagulation-diffusion PDE is a system of partial differential equations modelling the evolution in time of mass-bearing Brownian particles which are subject to short-range pairwise coagulation. This survey presents a fairly detailed exposition of the kinetic limit derivation of the Smoluchowski PDE from a microscopic model of many coagulating Brownian particles that was undertaken in [11]. It presents heuristic explanations of the form of the main theorem before discussing the proof, and presents key estimates in that proof using a novel probabilistic technique. The survey's p"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.0829","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":""},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:54:07Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"shX23Q7XgBf9+T8O7/P9INlwCMiM2JPfxcSCSPnBUkkh4VbQzPVJ/pBKsc3fQu4q1PL1gyEnfnO3dblMRP2fCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T07:23:24.174823Z"},"content_sha256":"578e832ab9121c17195f393222132422024c3f51f67b8e54bf1bc8c1cca5f496","schema_version":"1.0","event_id":"sha256:578e832ab9121c17195f393222132422024c3f51f67b8e54bf1bc8c1cca5f496"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/TDFNHFBKU7PPQGH5FED7HXT3CM/bundle.json","state_url":"https://pith.science/pith/TDFNHFBKU7PPQGH5FED7HXT3CM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/TDFNHFBKU7PPQGH5FED7HXT3CM/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-20T07:23:24Z","links":{"resolver":"https://pith.science/pith/TDFNHFBKU7PPQGH5FED7HXT3CM","bundle":"https://pith.science/pith/TDFNHFBKU7PPQGH5FED7HXT3CM/bundle.json","state":"https://pith.science/pith/TDFNHFBKU7PPQGH5FED7HXT3CM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/TDFNHFBKU7PPQGH5FED7HXT3CM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:TDFNHFBKU7PPQGH5FED7HXT3CM","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":"d0b29b8321bebd60eec168fbf7d214bb485b2552912a4ab2276137109aacd301","cross_cats_sorted":["math-ph","math.AP","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-01-04T17:19:14Z","title_canon_sha256":"1d3aa14fabfca478560478d70e2a83b4584bbced649d80207109b1bd9886460c"},"schema_version":"1.0","source":{"id":"1401.0829","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1401.0829","created_at":"2026-05-17T23:54:07Z"},{"alias_kind":"arxiv_version","alias_value":"1401.0829v2","created_at":"2026-05-17T23:54:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1401.0829","created_at":"2026-05-17T23:54:07Z"},{"alias_kind":"pith_short_12","alias_value":"TDFNHFBKU7PP","created_at":"2026-05-18T12:28:49Z"},{"alias_kind":"pith_short_16","alias_value":"TDFNHFBKU7PPQGH5","created_at":"2026-05-18T12:28:49Z"},{"alias_kind":"pith_short_8","alias_value":"TDFNHFBK","created_at":"2026-05-18T12:28:49Z"}],"graph_snapshots":[{"event_id":"sha256:578e832ab9121c17195f393222132422024c3f51f67b8e54bf1bc8c1cca5f496","target":"graph","created_at":"2026-05-17T23:54:07Z","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":"The Smoluchowski coagulation-diffusion PDE is a system of partial differential equations modelling the evolution in time of mass-bearing Brownian particles which are subject to short-range pairwise coagulation. This survey presents a fairly detailed exposition of the kinetic limit derivation of the Smoluchowski PDE from a microscopic model of many coagulating Brownian particles that was undertaken in [11]. It presents heuristic explanations of the form of the main theorem before discussing the proof, and presents key estimates in that proof using a novel probabilistic technique. The survey's p","authors_text":"Alan Hammond","cross_cats":["math-ph","math.AP","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-01-04T17:19:14Z","title":"Coagulation and diffusion: a probabilistic perspective on the Smoluchowski PDE"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.0829","kind":"arxiv","version":2},"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:ce3b31b9fb3f6aaade9464c883e51fb80b2922c635a26c45bc5f665f60dbe25c","target":"record","created_at":"2026-05-17T23:54:07Z","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":"d0b29b8321bebd60eec168fbf7d214bb485b2552912a4ab2276137109aacd301","cross_cats_sorted":["math-ph","math.AP","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-01-04T17:19:14Z","title_canon_sha256":"1d3aa14fabfca478560478d70e2a83b4584bbced649d80207109b1bd9886460c"},"schema_version":"1.0","source":{"id":"1401.0829","kind":"arxiv","version":2}},"canonical_sha256":"98cad3942aa7def818fd2907f3de7b132d378639c4258981a72583ac4a628ba6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"98cad3942aa7def818fd2907f3de7b132d378639c4258981a72583ac4a628ba6","first_computed_at":"2026-05-17T23:54:07.918691Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:54:07.918691Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"TaqXSUFmtb0x0HmUfzV5fTQKReo8ecj54YMpPjzM9hQ3hg/MI0FNqRkpU+fZwo3tt1ux/gqDQeQpfCnLvTDLBQ==","signature_status":"signed_v1","signed_at":"2026-05-17T23:54:07.919125Z","signed_message":"canonical_sha256_bytes"},"source_id":"1401.0829","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ce3b31b9fb3f6aaade9464c883e51fb80b2922c635a26c45bc5f665f60dbe25c","sha256:578e832ab9121c17195f393222132422024c3f51f67b8e54bf1bc8c1cca5f496"],"state_sha256":"b87bf1c93af8b11f83e8c6edb0cf678c8d873e53a6af0d965a5cab2023da7095"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"iQmkx1GFcrp4GUf74Q2BGI2NBexfo0DHBtBoKSRcrRBzAt/m2VIk9bMutxmd3+sRZrWRgj3IXulnRifAuGGvBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-20T07:23:24.176766Z","bundle_sha256":"e6e6bc7b4434ad99e9d210d0c72ba663d2b07b79a4e69e8578378b0a1bf60574"}}