{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:W45GCNI2ECUCJ5QAL4J6V3JUSQ","short_pith_number":"pith:W45GCNI2","canonical_record":{"source":{"id":"1412.5645","kind":"arxiv","version":11},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-12-17T22:06:26Z","cross_cats_sorted":[],"title_canon_sha256":"0dce91f2e6a176af221c9549375d9fb9e0f4c6b26b6d222bdacd28f38fb4de98","abstract_canon_sha256":"07aec61aaffbde3c40b73ac38d3168055537d93886fe5f9cba19ac387c7b31e9"},"schema_version":"1.0"},"canonical_sha256":"b73a61351a20a824f6005f13eaed3494325b12f776209080e3dc4b2813fc81da","source":{"kind":"arxiv","id":"1412.5645","version":11},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1412.5645","created_at":"2026-05-18T01:01:02Z"},{"alias_kind":"arxiv_version","alias_value":"1412.5645v11","created_at":"2026-05-18T01:01:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1412.5645","created_at":"2026-05-18T01:01:02Z"},{"alias_kind":"pith_short_12","alias_value":"W45GCNI2ECUC","created_at":"2026-05-18T12:28:54Z"},{"alias_kind":"pith_short_16","alias_value":"W45GCNI2ECUCJ5QA","created_at":"2026-05-18T12:28:54Z"},{"alias_kind":"pith_short_8","alias_value":"W45GCNI2","created_at":"2026-05-18T12:28:54Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:W45GCNI2ECUCJ5QAL4J6V3JUSQ","target":"record","payload":{"canonical_record":{"source":{"id":"1412.5645","kind":"arxiv","version":11},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-12-17T22:06:26Z","cross_cats_sorted":[],"title_canon_sha256":"0dce91f2e6a176af221c9549375d9fb9e0f4c6b26b6d222bdacd28f38fb4de98","abstract_canon_sha256":"07aec61aaffbde3c40b73ac38d3168055537d93886fe5f9cba19ac387c7b31e9"},"schema_version":"1.0"},"canonical_sha256":"b73a61351a20a824f6005f13eaed3494325b12f776209080e3dc4b2813fc81da","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:01:02.020052Z","signature_b64":"/XxXERBLvhJiJoxoft9/IhBQL7iEDR5q0DqkdqFtQayt/j53PXfdEiG1ngFPclCGu8k5X/izISFchx3+3iD7Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b73a61351a20a824f6005f13eaed3494325b12f776209080e3dc4b2813fc81da","last_reissued_at":"2026-05-18T01:01:02.019239Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:01:02.019239Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1412.5645","source_version":11,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:01:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IKixV5umuH5UoArZKDp1n2kkdbWWtr6b0n0nqd1y6oUCwYW0zRtLqry5rpRhixg9/X/rezSIw/59F0ymxTODBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T17:20:38.704592Z"},"content_sha256":"09de39e3dd9a33e70b26d864456c526b67c497a5546c8bd89f66ec516190c8dd","schema_version":"1.0","event_id":"sha256:09de39e3dd9a33e70b26d864456c526b67c497a5546c8bd89f66ec516190c8dd"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:W45GCNI2ECUCJ5QAL4J6V3JUSQ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Effects of microscopic dynamics on Brownian coagulation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Guolong Li","submitted_at":"2014-12-17T22:06:26Z","abstract_excerpt":"We consider two different models for colloidal particles. In the first model, we consider their free motion to be diffusion while in the second model we take it to be integrated Ornstein-Uhlenbeck process. In both models, we derived collision estimates for pairs of particles. In particular, we found that these estimates would be different to the Brownian case even when the particles' free motion is Brownian at macroscopic scales. As a consequence, the coagulation kernel and diffusivity in the coagulation-diffusion equations would also be affected accordingly. We then proved that there exists a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.5645","kind":"arxiv","version":11},"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":"2026-05-18T01:01:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wb3laiDIkw65zWx5o1o5rB5FV5/lr7jdDV39GwTg5CVtqNycncEbY6dD7ky+CnTDjp8NZW22PQnH+dmnCWHiAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T17:20:38.704930Z"},"content_sha256":"742b3ec83106bc845b8b8309b06ace110207e5f8903c357a2747335cba578ae8","schema_version":"1.0","event_id":"sha256:742b3ec83106bc845b8b8309b06ace110207e5f8903c357a2747335cba578ae8"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/W45GCNI2ECUCJ5QAL4J6V3JUSQ/bundle.json","state_url":"https://pith.science/pith/W45GCNI2ECUCJ5QAL4J6V3JUSQ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/W45GCNI2ECUCJ5QAL4J6V3JUSQ/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":"2026-06-27T17:20:38Z","links":{"resolver":"https://pith.science/pith/W45GCNI2ECUCJ5QAL4J6V3JUSQ","bundle":"https://pith.science/pith/W45GCNI2ECUCJ5QAL4J6V3JUSQ/bundle.json","state":"https://pith.science/pith/W45GCNI2ECUCJ5QAL4J6V3JUSQ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/W45GCNI2ECUCJ5QAL4J6V3JUSQ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:W45GCNI2ECUCJ5QAL4J6V3JUSQ","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":"07aec61aaffbde3c40b73ac38d3168055537d93886fe5f9cba19ac387c7b31e9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-12-17T22:06:26Z","title_canon_sha256":"0dce91f2e6a176af221c9549375d9fb9e0f4c6b26b6d222bdacd28f38fb4de98"},"schema_version":"1.0","source":{"id":"1412.5645","kind":"arxiv","version":11}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1412.5645","created_at":"2026-05-18T01:01:02Z"},{"alias_kind":"arxiv_version","alias_value":"1412.5645v11","created_at":"2026-05-18T01:01:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1412.5645","created_at":"2026-05-18T01:01:02Z"},{"alias_kind":"pith_short_12","alias_value":"W45GCNI2ECUC","created_at":"2026-05-18T12:28:54Z"},{"alias_kind":"pith_short_16","alias_value":"W45GCNI2ECUCJ5QA","created_at":"2026-05-18T12:28:54Z"},{"alias_kind":"pith_short_8","alias_value":"W45GCNI2","created_at":"2026-05-18T12:28:54Z"}],"graph_snapshots":[{"event_id":"sha256:742b3ec83106bc845b8b8309b06ace110207e5f8903c357a2747335cba578ae8","target":"graph","created_at":"2026-05-18T01:01:02Z","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":"We consider two different models for colloidal particles. In the first model, we consider their free motion to be diffusion while in the second model we take it to be integrated Ornstein-Uhlenbeck process. In both models, we derived collision estimates for pairs of particles. In particular, we found that these estimates would be different to the Brownian case even when the particles' free motion is Brownian at macroscopic scales. As a consequence, the coagulation kernel and diffusivity in the coagulation-diffusion equations would also be affected accordingly. We then proved that there exists a","authors_text":"Guolong Li","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-12-17T22:06:26Z","title":"Effects of microscopic dynamics on Brownian coagulation"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.5645","kind":"arxiv","version":11},"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":"sha256:09de39e3dd9a33e70b26d864456c526b67c497a5546c8bd89f66ec516190c8dd","target":"record","created_at":"2026-05-18T01:01:02Z","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":"07aec61aaffbde3c40b73ac38d3168055537d93886fe5f9cba19ac387c7b31e9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-12-17T22:06:26Z","title_canon_sha256":"0dce91f2e6a176af221c9549375d9fb9e0f4c6b26b6d222bdacd28f38fb4de98"},"schema_version":"1.0","source":{"id":"1412.5645","kind":"arxiv","version":11}},"canonical_sha256":"b73a61351a20a824f6005f13eaed3494325b12f776209080e3dc4b2813fc81da","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b73a61351a20a824f6005f13eaed3494325b12f776209080e3dc4b2813fc81da","first_computed_at":"2026-05-18T01:01:02.019239Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:01:02.019239Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"/XxXERBLvhJiJoxoft9/IhBQL7iEDR5q0DqkdqFtQayt/j53PXfdEiG1ngFPclCGu8k5X/izISFchx3+3iD7Cg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:01:02.020052Z","signed_message":"canonical_sha256_bytes"},"source_id":"1412.5645","source_kind":"arxiv","source_version":11}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:09de39e3dd9a33e70b26d864456c526b67c497a5546c8bd89f66ec516190c8dd","sha256:742b3ec83106bc845b8b8309b06ace110207e5f8903c357a2747335cba578ae8"],"state_sha256":"80064519d8c562ef5686233a9ea003c64c07906ba7a9dbac8b0faebd2487f589"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BBFeMpKh0s5eiTw3/g0MFfoT+1q+M5vJSIlqhNXem6qgiAq+DMhPn7CKbTJ7EKmN4eXaPA25D995v4tDVBN+AQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-27T17:20:38.706816Z","bundle_sha256":"3b4aaeb133370c42f100724e87919413bf98a7a3e60db20691ed8000bb67f050"}}