{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2008:FOJQ53JUPL4YJYFBQPDTWZDE7C","short_pith_number":"pith:FOJQ53JU","canonical_record":{"source":{"id":"0808.1064","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2008-08-07T16:17:23Z","cross_cats_sorted":["math.FA","math.MP"],"title_canon_sha256":"b4a52a285d943e27a507197d1347d2a409b1a424eb0629a1ce9445ead8e51fc2","abstract_canon_sha256":"4cf16a7d1238fa24b44975ff609adaead292159ff1c279426ef395688204fc39"},"schema_version":"1.0"},"canonical_sha256":"2b930eed347af984e0a183c73b6464f8a70234d5144308f77d01f9babb602fe6","source":{"kind":"arxiv","id":"0808.1064","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0808.1064","created_at":"2026-05-18T02:15:43Z"},{"alias_kind":"arxiv_version","alias_value":"0808.1064v2","created_at":"2026-05-18T02:15:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0808.1064","created_at":"2026-05-18T02:15:43Z"},{"alias_kind":"pith_short_12","alias_value":"FOJQ53JUPL4Y","created_at":"2026-05-18T12:25:57Z"},{"alias_kind":"pith_short_16","alias_value":"FOJQ53JUPL4YJYFB","created_at":"2026-05-18T12:25:57Z"},{"alias_kind":"pith_short_8","alias_value":"FOJQ53JU","created_at":"2026-05-18T12:25:57Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2008:FOJQ53JUPL4YJYFBQPDTWZDE7C","target":"record","payload":{"canonical_record":{"source":{"id":"0808.1064","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2008-08-07T16:17:23Z","cross_cats_sorted":["math.FA","math.MP"],"title_canon_sha256":"b4a52a285d943e27a507197d1347d2a409b1a424eb0629a1ce9445ead8e51fc2","abstract_canon_sha256":"4cf16a7d1238fa24b44975ff609adaead292159ff1c279426ef395688204fc39"},"schema_version":"1.0"},"canonical_sha256":"2b930eed347af984e0a183c73b6464f8a70234d5144308f77d01f9babb602fe6","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:15:43.785775Z","signature_b64":"BKeHYNqKpYkXP58zeakFua7yYH94ZywpNF9LV2fEzw2MnHHBPsv/3s+g+ZvkdXnmb0dD4m4BfpP765LiJDspDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2b930eed347af984e0a183c73b6464f8a70234d5144308f77d01f9babb602fe6","last_reissued_at":"2026-05-18T02:15:43.785045Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:15:43.785045Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0808.1064","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":"2026-05-18T02:15:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"MVfCGUGbULpLzIeMQWUygpDIXY0BLFm21QPtBhn+RZLcYalBYA1rrEPz6B+Yy43dby+lAfNqa7oa1LunyZR6Ag==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T21:20:31.066751Z"},"content_sha256":"22b8662b1298a326d3add8965edea7f31c04985d2e9f4fa3313fa6bc0746c98e","schema_version":"1.0","event_id":"sha256:22b8662b1298a326d3add8965edea7f31c04985d2e9f4fa3313fa6bc0746c98e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2008:FOJQ53JUPL4YJYFBQPDTWZDE7C","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On Strong Convergence to Equilibrium for the Boltzmann Equation with Soft Potentials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA","math.MP"],"primary_cat":"math-ph","authors_text":"Eric A. Carlen, Maria C. Carvalho, Xuguang Lu","submitted_at":"2008-08-07T16:17:23Z","abstract_excerpt":"The paper concerns $L^1$- convergence to equilibrium for weak solutions of the spatially homogeneous Boltzmann Equation for soft potentials $(-4\\le \\gm<0$), with and without angular cutoff. We prove the time-averaged $L^1$-convergence to equilibrium for all weak solutions whose initial data have finite entropy and finite moments up to order greater than $2+|\\gm|$. For the usual $L^1$-convergence we prove that the convergence rate can be controlled from below by the initial energy tails, and hence, for initial data with long energy tails, the convergence can be arbitrarily slow. We also show th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0808.1064","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":"2026-05-18T02:15:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xxFvUeaKe+wa4sWIZnHmzcVDwa2Uqro7INoQIueps2FdZPMiJZObzocFP3YbAMeAYTToA8fNag6P6ycJ8XF9AQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T21:20:31.067094Z"},"content_sha256":"2eaebd83558b9096a573cc33583554f96c47149af2d120fba3b4e2ba2e6073ed","schema_version":"1.0","event_id":"sha256:2eaebd83558b9096a573cc33583554f96c47149af2d120fba3b4e2ba2e6073ed"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/FOJQ53JUPL4YJYFBQPDTWZDE7C/bundle.json","state_url":"https://pith.science/pith/FOJQ53JUPL4YJYFBQPDTWZDE7C/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/FOJQ53JUPL4YJYFBQPDTWZDE7C/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":"2026-06-23T21:20:31Z","links":{"resolver":"https://pith.science/pith/FOJQ53JUPL4YJYFBQPDTWZDE7C","bundle":"https://pith.science/pith/FOJQ53JUPL4YJYFBQPDTWZDE7C/bundle.json","state":"https://pith.science/pith/FOJQ53JUPL4YJYFBQPDTWZDE7C/state.json","well_known_bundle":"https://pith.science/.well-known/pith/FOJQ53JUPL4YJYFBQPDTWZDE7C/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2008:FOJQ53JUPL4YJYFBQPDTWZDE7C","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":"4cf16a7d1238fa24b44975ff609adaead292159ff1c279426ef395688204fc39","cross_cats_sorted":["math.FA","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2008-08-07T16:17:23Z","title_canon_sha256":"b4a52a285d943e27a507197d1347d2a409b1a424eb0629a1ce9445ead8e51fc2"},"schema_version":"1.0","source":{"id":"0808.1064","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0808.1064","created_at":"2026-05-18T02:15:43Z"},{"alias_kind":"arxiv_version","alias_value":"0808.1064v2","created_at":"2026-05-18T02:15:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0808.1064","created_at":"2026-05-18T02:15:43Z"},{"alias_kind":"pith_short_12","alias_value":"FOJQ53JUPL4Y","created_at":"2026-05-18T12:25:57Z"},{"alias_kind":"pith_short_16","alias_value":"FOJQ53JUPL4YJYFB","created_at":"2026-05-18T12:25:57Z"},{"alias_kind":"pith_short_8","alias_value":"FOJQ53JU","created_at":"2026-05-18T12:25:57Z"}],"graph_snapshots":[{"event_id":"sha256:2eaebd83558b9096a573cc33583554f96c47149af2d120fba3b4e2ba2e6073ed","target":"graph","created_at":"2026-05-18T02:15:43Z","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 paper concerns $L^1$- convergence to equilibrium for weak solutions of the spatially homogeneous Boltzmann Equation for soft potentials $(-4\\le \\gm<0$), with and without angular cutoff. We prove the time-averaged $L^1$-convergence to equilibrium for all weak solutions whose initial data have finite entropy and finite moments up to order greater than $2+|\\gm|$. For the usual $L^1$-convergence we prove that the convergence rate can be controlled from below by the initial energy tails, and hence, for initial data with long energy tails, the convergence can be arbitrarily slow. We also show th","authors_text":"Eric A. Carlen, Maria C. Carvalho, Xuguang Lu","cross_cats":["math.FA","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2008-08-07T16:17:23Z","title":"On Strong Convergence to Equilibrium for the Boltzmann Equation with Soft Potentials"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0808.1064","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":"sha256:22b8662b1298a326d3add8965edea7f31c04985d2e9f4fa3313fa6bc0746c98e","target":"record","created_at":"2026-05-18T02:15:43Z","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":"4cf16a7d1238fa24b44975ff609adaead292159ff1c279426ef395688204fc39","cross_cats_sorted":["math.FA","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2008-08-07T16:17:23Z","title_canon_sha256":"b4a52a285d943e27a507197d1347d2a409b1a424eb0629a1ce9445ead8e51fc2"},"schema_version":"1.0","source":{"id":"0808.1064","kind":"arxiv","version":2}},"canonical_sha256":"2b930eed347af984e0a183c73b6464f8a70234d5144308f77d01f9babb602fe6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2b930eed347af984e0a183c73b6464f8a70234d5144308f77d01f9babb602fe6","first_computed_at":"2026-05-18T02:15:43.785045Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:15:43.785045Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"BKeHYNqKpYkXP58zeakFua7yYH94ZywpNF9LV2fEzw2MnHHBPsv/3s+g+ZvkdXnmb0dD4m4BfpP765LiJDspDw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:15:43.785775Z","signed_message":"canonical_sha256_bytes"},"source_id":"0808.1064","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:22b8662b1298a326d3add8965edea7f31c04985d2e9f4fa3313fa6bc0746c98e","sha256:2eaebd83558b9096a573cc33583554f96c47149af2d120fba3b4e2ba2e6073ed"],"state_sha256":"399ab4b6049a13f22185fa73a1c498bf307186d0cc53b0c46c09d8d03d346710"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZzkOgkkB5N9lUHtFa3v+g+5w9pBA53JBpW4pjrZZ+FSTmYORL4Llq3poTO+ef3Bzh52B5ZcrSL2ny7N0faCMAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T21:20:31.069051Z","bundle_sha256":"8444ebdff6f87b8b4c4d4c77f68147bfb8bab5b3627b1b9376468c146b2a04d3"}}