{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:WGNIYM7UXKCCS5YUZXQEGBO2N3","short_pith_number":"pith:WGNIYM7U","canonical_record":{"source":{"id":"1102.5291","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cond-mat.stat-mech","submitted_at":"2011-02-25T17:23:47Z","cross_cats_sorted":[],"title_canon_sha256":"dad0faadca3a9ea2d3445df7b6704cb44e1a8d8e755169baeda8a699cd88da6a","abstract_canon_sha256":"8a6bf75861c2717512945fd6dbde4dbe67800897d1573a156c554d3581090e58"},"schema_version":"1.0"},"canonical_sha256":"b19a8c33f4ba84297714cde04305da6ee1aa7e3a2ed13a9bc4e9db4bf3885f89","source":{"kind":"arxiv","id":"1102.5291","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1102.5291","created_at":"2026-05-18T04:18:54Z"},{"alias_kind":"arxiv_version","alias_value":"1102.5291v1","created_at":"2026-05-18T04:18:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1102.5291","created_at":"2026-05-18T04:18:54Z"},{"alias_kind":"pith_short_12","alias_value":"WGNIYM7UXKCC","created_at":"2026-05-18T12:26:44Z"},{"alias_kind":"pith_short_16","alias_value":"WGNIYM7UXKCCS5YU","created_at":"2026-05-18T12:26:44Z"},{"alias_kind":"pith_short_8","alias_value":"WGNIYM7U","created_at":"2026-05-18T12:26:44Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:WGNIYM7UXKCCS5YUZXQEGBO2N3","target":"record","payload":{"canonical_record":{"source":{"id":"1102.5291","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cond-mat.stat-mech","submitted_at":"2011-02-25T17:23:47Z","cross_cats_sorted":[],"title_canon_sha256":"dad0faadca3a9ea2d3445df7b6704cb44e1a8d8e755169baeda8a699cd88da6a","abstract_canon_sha256":"8a6bf75861c2717512945fd6dbde4dbe67800897d1573a156c554d3581090e58"},"schema_version":"1.0"},"canonical_sha256":"b19a8c33f4ba84297714cde04305da6ee1aa7e3a2ed13a9bc4e9db4bf3885f89","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:18:54.490534Z","signature_b64":"t99YjkhItWT5a5yATKiIFvmj/pUX/dRfcJv5x7jec7zM9L/2Pl+caVO2+zAWjSys6ep92PhyiC8t9fQk75itBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b19a8c33f4ba84297714cde04305da6ee1aa7e3a2ed13a9bc4e9db4bf3885f89","last_reissued_at":"2026-05-18T04:18:54.490155Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:18:54.490155Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1102.5291","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:18:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AwJoPihi7wbpo5RIzlzF+ui0/lsBxTuArZ75cc1L2XV6dOhkC1nrvs8mG9zC4jHJALLOvPf3/aREISl2N3C4BQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T03:23:08.238966Z"},"content_sha256":"c5f61d60a15246a211a3270748936e22c5ec61486a8db76bb4ce32f4184a5131","schema_version":"1.0","event_id":"sha256:c5f61d60a15246a211a3270748936e22c5ec61486a8db76bb4ce32f4184a5131"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:WGNIYM7UXKCCS5YUZXQEGBO2N3","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Collision number statistics for transport processes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cond-mat.stat-mech","authors_text":"Alain Mazzolo, Andrea Zoia, Eric Dumonteil","submitted_at":"2011-02-25T17:23:47Z","abstract_excerpt":"Many physical observables can be represented as a particle spending some random time within a given domain. For a broad class of transport-dominated processes, we detail how it is possible to express the moments of the number of particle collisions in an arbitrary volume in terms of repeated convolutions of the ensemble equilibrium distribution. This approach is shown to generalize the celebrated Kac formula for the moments of residence times, which is recovered in the diffusion limit. Some practical applications are illustrated for bounded, unbounded and absorbing domains."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1102.5291","kind":"arxiv","version":1},"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":"2026-05-18T04:18:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JCQN2F5SpKGhvvF1TzNLG6pqnnOSThyHaREiC1ViT7vPyqxh1cpz75k/LoQuf3SdPH5HH/3PA6AIo3ya5jQLAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T03:23:08.239326Z"},"content_sha256":"449f4d7e5bec2a672f8f83572de9339b4c3bb4375a53c30ab523e41103417f20","schema_version":"1.0","event_id":"sha256:449f4d7e5bec2a672f8f83572de9339b4c3bb4375a53c30ab523e41103417f20"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/WGNIYM7UXKCCS5YUZXQEGBO2N3/bundle.json","state_url":"https://pith.science/pith/WGNIYM7UXKCCS5YUZXQEGBO2N3/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/WGNIYM7UXKCCS5YUZXQEGBO2N3/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":"2026-06-03T03:23:08Z","links":{"resolver":"https://pith.science/pith/WGNIYM7UXKCCS5YUZXQEGBO2N3","bundle":"https://pith.science/pith/WGNIYM7UXKCCS5YUZXQEGBO2N3/bundle.json","state":"https://pith.science/pith/WGNIYM7UXKCCS5YUZXQEGBO2N3/state.json","well_known_bundle":"https://pith.science/.well-known/pith/WGNIYM7UXKCCS5YUZXQEGBO2N3/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:WGNIYM7UXKCCS5YUZXQEGBO2N3","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":"8a6bf75861c2717512945fd6dbde4dbe67800897d1573a156c554d3581090e58","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cond-mat.stat-mech","submitted_at":"2011-02-25T17:23:47Z","title_canon_sha256":"dad0faadca3a9ea2d3445df7b6704cb44e1a8d8e755169baeda8a699cd88da6a"},"schema_version":"1.0","source":{"id":"1102.5291","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1102.5291","created_at":"2026-05-18T04:18:54Z"},{"alias_kind":"arxiv_version","alias_value":"1102.5291v1","created_at":"2026-05-18T04:18:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1102.5291","created_at":"2026-05-18T04:18:54Z"},{"alias_kind":"pith_short_12","alias_value":"WGNIYM7UXKCC","created_at":"2026-05-18T12:26:44Z"},{"alias_kind":"pith_short_16","alias_value":"WGNIYM7UXKCCS5YU","created_at":"2026-05-18T12:26:44Z"},{"alias_kind":"pith_short_8","alias_value":"WGNIYM7U","created_at":"2026-05-18T12:26:44Z"}],"graph_snapshots":[{"event_id":"sha256:449f4d7e5bec2a672f8f83572de9339b4c3bb4375a53c30ab523e41103417f20","target":"graph","created_at":"2026-05-18T04:18:54Z","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":"Many physical observables can be represented as a particle spending some random time within a given domain. For a broad class of transport-dominated processes, we detail how it is possible to express the moments of the number of particle collisions in an arbitrary volume in terms of repeated convolutions of the ensemble equilibrium distribution. This approach is shown to generalize the celebrated Kac formula for the moments of residence times, which is recovered in the diffusion limit. Some practical applications are illustrated for bounded, unbounded and absorbing domains.","authors_text":"Alain Mazzolo, Andrea Zoia, Eric Dumonteil","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cond-mat.stat-mech","submitted_at":"2011-02-25T17:23:47Z","title":"Collision number statistics for transport processes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1102.5291","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:c5f61d60a15246a211a3270748936e22c5ec61486a8db76bb4ce32f4184a5131","target":"record","created_at":"2026-05-18T04:18:54Z","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":"8a6bf75861c2717512945fd6dbde4dbe67800897d1573a156c554d3581090e58","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cond-mat.stat-mech","submitted_at":"2011-02-25T17:23:47Z","title_canon_sha256":"dad0faadca3a9ea2d3445df7b6704cb44e1a8d8e755169baeda8a699cd88da6a"},"schema_version":"1.0","source":{"id":"1102.5291","kind":"arxiv","version":1}},"canonical_sha256":"b19a8c33f4ba84297714cde04305da6ee1aa7e3a2ed13a9bc4e9db4bf3885f89","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b19a8c33f4ba84297714cde04305da6ee1aa7e3a2ed13a9bc4e9db4bf3885f89","first_computed_at":"2026-05-18T04:18:54.490155Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:18:54.490155Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"t99YjkhItWT5a5yATKiIFvmj/pUX/dRfcJv5x7jec7zM9L/2Pl+caVO2+zAWjSys6ep92PhyiC8t9fQk75itBw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:18:54.490534Z","signed_message":"canonical_sha256_bytes"},"source_id":"1102.5291","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c5f61d60a15246a211a3270748936e22c5ec61486a8db76bb4ce32f4184a5131","sha256:449f4d7e5bec2a672f8f83572de9339b4c3bb4375a53c30ab523e41103417f20"],"state_sha256":"84e7379ab0c51f86b3a4a4d42462913e6d9cfbecd90adc459514e508388997e8"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AGhLo/MK0mjMJ2XVVnEGzD0ZsV16gYSimNfPETLbXwjx0BJz1QbVi7z0nHd6RmLnqVXaX4+/WJWtfettiVroCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T03:23:08.241327Z","bundle_sha256":"22c32f5455f4da280d64155e6967ca1f633a9c7698967b179520db523cb19577"}}