{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2005:NXC43WTAYNW347HV2GT3PKRJPC","short_pith_number":"pith:NXC43WTA","canonical_record":{"source":{"id":"nlin/0507017","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"nlin.CD","submitted_at":"2005-07-08T16:26:18Z","cross_cats_sorted":["cond-mat.stat-mech"],"title_canon_sha256":"a69e2222be507d2fce229d85152c14197fba9b7626a52c0cddbc1b03eceda9a1","abstract_canon_sha256":"e22e957d5cecef67bd71891709d3655f4194905dc76609ff288040ae3f8b1bbf"},"schema_version":"1.0"},"canonical_sha256":"6dc5cdda60c36dbe7cf5d1a7b7aa2978aec1e2cf3c9aec871a77b1852bca10ac","source":{"kind":"arxiv","id":"nlin/0507017","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"nlin/0507017","created_at":"2026-05-18T01:05:21Z"},{"alias_kind":"arxiv_version","alias_value":"nlin/0507017v2","created_at":"2026-05-18T01:05:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.nlin/0507017","created_at":"2026-05-18T01:05:21Z"},{"alias_kind":"pith_short_12","alias_value":"NXC43WTAYNW3","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_16","alias_value":"NXC43WTAYNW347HV","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_8","alias_value":"NXC43WTA","created_at":"2026-05-18T12:25:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2005:NXC43WTAYNW347HV2GT3PKRJPC","target":"record","payload":{"canonical_record":{"source":{"id":"nlin/0507017","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"nlin.CD","submitted_at":"2005-07-08T16:26:18Z","cross_cats_sorted":["cond-mat.stat-mech"],"title_canon_sha256":"a69e2222be507d2fce229d85152c14197fba9b7626a52c0cddbc1b03eceda9a1","abstract_canon_sha256":"e22e957d5cecef67bd71891709d3655f4194905dc76609ff288040ae3f8b1bbf"},"schema_version":"1.0"},"canonical_sha256":"6dc5cdda60c36dbe7cf5d1a7b7aa2978aec1e2cf3c9aec871a77b1852bca10ac","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:05:21.736938Z","signature_b64":"q7RRjE2PyrtKouvpn8t7GJyY8rToD8cKg7HRIxd5QHe4HsQoMC4XYIwJ4JhEWioxxB+K+QnOqJ6SRfoLqJWGDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6dc5cdda60c36dbe7cf5d1a7b7aa2978aec1e2cf3c9aec871a77b1852bca10ac","last_reissued_at":"2026-05-18T01:05:21.736346Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:05:21.736346Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"nlin/0507017","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":"2026-05-18T01:05:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tiG30RVAdwJblPXOk9DztkTHIjVrU3GG7NR2lzWD1hS1d61piKyEk4bkQmiI/Mb4KOEqKQIz6jrD3kh+yJIXBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T22:05:32.820127Z"},"content_sha256":"f3d7ed0643609ed741f2093a18c8171af24ffef7cedd6a632daa262b5c38bcef","schema_version":"1.0","event_id":"sha256:f3d7ed0643609ed741f2093a18c8171af24ffef7cedd6a632daa262b5c38bcef"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2005:NXC43WTAYNW347HV2GT3PKRJPC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Systematic Density Expansion of the Lyapunov Exponents for a Two-dimensional Random Lorentz Gas","license":"","headline":"","cross_cats":["cond-mat.stat-mech"],"primary_cat":"nlin.CD","authors_text":"Debabrata Panja, Henk van Beijeren, H. V. Kruis","submitted_at":"2005-07-08T16:26:18Z","abstract_excerpt":"We study the Lyapunov exponents of a two-dimensional, random Lorentz gas at low density. The positive Lyapunov exponent may be obtained either by a direct analysis of the dynamics, or by the use of kinetic theory methods. To leading orders in the density of scatterers it is of the form $A_{0}\\tilde{n}\\ln\\tilde{n}+B_{0}\\tilde{n}$, where $A_{0}$ and $B_{0}$ are known constants and $\\tilde{n}$ is the number density of scatterers expressed in dimensionless units. In this paper, we find that through order $(\\tilde{n}^{2})$, the positive Lyapunov exponent is of the form $A_{0}\\tilde{n}\\ln\\tilde{n}+B"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"nlin/0507017","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":"2026-05-18T01:05:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+K19mwhQFuSHQauqB6X8JTyoBxkjB5eQWfteidRQv0Klb6v/gjFCaqFmZffQIWWBzecKHoFxK1BwS9IMoRNvDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T22:05:32.820819Z"},"content_sha256":"05461315e1a8685780ab5b9abf67581b8742f7b4fbf7f190c0889a96892fd400","schema_version":"1.0","event_id":"sha256:05461315e1a8685780ab5b9abf67581b8742f7b4fbf7f190c0889a96892fd400"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/NXC43WTAYNW347HV2GT3PKRJPC/bundle.json","state_url":"https://pith.science/pith/NXC43WTAYNW347HV2GT3PKRJPC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/NXC43WTAYNW347HV2GT3PKRJPC/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":"2026-05-25T22:05:32Z","links":{"resolver":"https://pith.science/pith/NXC43WTAYNW347HV2GT3PKRJPC","bundle":"https://pith.science/pith/NXC43WTAYNW347HV2GT3PKRJPC/bundle.json","state":"https://pith.science/pith/NXC43WTAYNW347HV2GT3PKRJPC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/NXC43WTAYNW347HV2GT3PKRJPC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2005:NXC43WTAYNW347HV2GT3PKRJPC","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":"e22e957d5cecef67bd71891709d3655f4194905dc76609ff288040ae3f8b1bbf","cross_cats_sorted":["cond-mat.stat-mech"],"license":"","primary_cat":"nlin.CD","submitted_at":"2005-07-08T16:26:18Z","title_canon_sha256":"a69e2222be507d2fce229d85152c14197fba9b7626a52c0cddbc1b03eceda9a1"},"schema_version":"1.0","source":{"id":"nlin/0507017","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"nlin/0507017","created_at":"2026-05-18T01:05:21Z"},{"alias_kind":"arxiv_version","alias_value":"nlin/0507017v2","created_at":"2026-05-18T01:05:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.nlin/0507017","created_at":"2026-05-18T01:05:21Z"},{"alias_kind":"pith_short_12","alias_value":"NXC43WTAYNW3","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_16","alias_value":"NXC43WTAYNW347HV","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_8","alias_value":"NXC43WTA","created_at":"2026-05-18T12:25:53Z"}],"graph_snapshots":[{"event_id":"sha256:05461315e1a8685780ab5b9abf67581b8742f7b4fbf7f190c0889a96892fd400","target":"graph","created_at":"2026-05-18T01:05:21Z","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 study the Lyapunov exponents of a two-dimensional, random Lorentz gas at low density. The positive Lyapunov exponent may be obtained either by a direct analysis of the dynamics, or by the use of kinetic theory methods. To leading orders in the density of scatterers it is of the form $A_{0}\\tilde{n}\\ln\\tilde{n}+B_{0}\\tilde{n}$, where $A_{0}$ and $B_{0}$ are known constants and $\\tilde{n}$ is the number density of scatterers expressed in dimensionless units. In this paper, we find that through order $(\\tilde{n}^{2})$, the positive Lyapunov exponent is of the form $A_{0}\\tilde{n}\\ln\\tilde{n}+B","authors_text":"Debabrata Panja, Henk van Beijeren, H. V. Kruis","cross_cats":["cond-mat.stat-mech"],"headline":"","license":"","primary_cat":"nlin.CD","submitted_at":"2005-07-08T16:26:18Z","title":"Systematic Density Expansion of the Lyapunov Exponents for a Two-dimensional Random Lorentz Gas"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"nlin/0507017","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:f3d7ed0643609ed741f2093a18c8171af24ffef7cedd6a632daa262b5c38bcef","target":"record","created_at":"2026-05-18T01:05:21Z","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":"e22e957d5cecef67bd71891709d3655f4194905dc76609ff288040ae3f8b1bbf","cross_cats_sorted":["cond-mat.stat-mech"],"license":"","primary_cat":"nlin.CD","submitted_at":"2005-07-08T16:26:18Z","title_canon_sha256":"a69e2222be507d2fce229d85152c14197fba9b7626a52c0cddbc1b03eceda9a1"},"schema_version":"1.0","source":{"id":"nlin/0507017","kind":"arxiv","version":2}},"canonical_sha256":"6dc5cdda60c36dbe7cf5d1a7b7aa2978aec1e2cf3c9aec871a77b1852bca10ac","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6dc5cdda60c36dbe7cf5d1a7b7aa2978aec1e2cf3c9aec871a77b1852bca10ac","first_computed_at":"2026-05-18T01:05:21.736346Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:05:21.736346Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"q7RRjE2PyrtKouvpn8t7GJyY8rToD8cKg7HRIxd5QHe4HsQoMC4XYIwJ4JhEWioxxB+K+QnOqJ6SRfoLqJWGDw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:05:21.736938Z","signed_message":"canonical_sha256_bytes"},"source_id":"nlin/0507017","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f3d7ed0643609ed741f2093a18c8171af24ffef7cedd6a632daa262b5c38bcef","sha256:05461315e1a8685780ab5b9abf67581b8742f7b4fbf7f190c0889a96892fd400"],"state_sha256":"84f3d1fc9615892454c39537a0b4de8f9fd906194295db146969a111bfd4e6d6"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9jTPXOUH8Ou44WTv0pDXbr0D2jbyUb2LaWvAXJQEoCQuUvw0KS+SRCFIWoQBANXu6Ay3Cvf5wtd7FAQC4xi6DA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-25T22:05:32.824190Z","bundle_sha256":"9625b05c9d678c3d8c8d3e96c11d41b7163d476c9a4e57ef5419d2a3d8ed6a0f"}}