{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:NJFRIA3Z7CEEHJ4QCG33ACZYPT","short_pith_number":"pith:NJFRIA3Z","canonical_record":{"source":{"id":"1505.01212","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-05-05T22:46:14Z","cross_cats_sorted":["math-ph","math.MP","math.PR"],"title_canon_sha256":"731434e7fafd7641b9d31a0d1b7a29ad63616debf4cd2c8656fc00c9c39ca86f","abstract_canon_sha256":"874a0b0f00ed09f3525436e65a2ea0ad837107bb51956f04931bc81bfba91bee"},"schema_version":"1.0"},"canonical_sha256":"6a4b140379f88843a79011b7b00b387ce793e1486e31e95b4f12ffde03268d85","source":{"kind":"arxiv","id":"1505.01212","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1505.01212","created_at":"2026-05-18T01:35:38Z"},{"alias_kind":"arxiv_version","alias_value":"1505.01212v3","created_at":"2026-05-18T01:35:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1505.01212","created_at":"2026-05-18T01:35:38Z"},{"alias_kind":"pith_short_12","alias_value":"NJFRIA3Z7CEE","created_at":"2026-05-18T12:29:32Z"},{"alias_kind":"pith_short_16","alias_value":"NJFRIA3Z7CEEHJ4Q","created_at":"2026-05-18T12:29:32Z"},{"alias_kind":"pith_short_8","alias_value":"NJFRIA3Z","created_at":"2026-05-18T12:29:32Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:NJFRIA3Z7CEEHJ4QCG33ACZYPT","target":"record","payload":{"canonical_record":{"source":{"id":"1505.01212","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-05-05T22:46:14Z","cross_cats_sorted":["math-ph","math.MP","math.PR"],"title_canon_sha256":"731434e7fafd7641b9d31a0d1b7a29ad63616debf4cd2c8656fc00c9c39ca86f","abstract_canon_sha256":"874a0b0f00ed09f3525436e65a2ea0ad837107bb51956f04931bc81bfba91bee"},"schema_version":"1.0"},"canonical_sha256":"6a4b140379f88843a79011b7b00b387ce793e1486e31e95b4f12ffde03268d85","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:35:38.049456Z","signature_b64":"dOFVG38jFsEusRwZ8b+f/OsRScor0ruCsDQohy3maavGj5R5Dsls0gJJkQ6GkOpJgTbzfx+Usn4V1maSfibnAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6a4b140379f88843a79011b7b00b387ce793e1486e31e95b4f12ffde03268d85","last_reissued_at":"2026-05-18T01:35:38.048836Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:35:38.048836Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1505.01212","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:35:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BOIf0+R79YwyedTAjGsh+DVmpaROUiS4cMaPT5XcwGTzBwpp+UMo5dwSqUBJvOdlFhbfqLrNaIZPQZpvz07vCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T13:30:30.871221Z"},"content_sha256":"289dd3f965c25e389cda32b231115b59ba0f42dd513e690a399e9cf2f497f4f6","schema_version":"1.0","event_id":"sha256:289dd3f965c25e389cda32b231115b59ba0f42dd513e690a399e9cf2f497f4f6"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:NJFRIA3Z7CEEHJ4QCG33ACZYPT","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Stationary solutions of the Vlasov-Fokker-Planck equation: existence, characterization and phase-transition","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP","math.PR"],"primary_cat":"math.AP","authors_text":"Julian Tugaut, Manh Hong Duong","submitted_at":"2015-05-05T22:46:14Z","abstract_excerpt":"In this paper, we study the set of stationary solutions of the Vlasov-Fokker-Planck (VFP) equation. This equation describes the time evolution of the probability distribution of a particle moving under the influence of a double-well potential, an interaction potential, a friction force and a stochastic force. We prove, under suitable assumptions, that the VFP equation does not have a unique stationary solution and that there exists a phase transition. Our study relies on the recent results by Tugaut and coauthors regarding the McKean-Vlasov equation."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.01212","kind":"arxiv","version":3},"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":"2026-05-18T01:35:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GClYCeem6RumhI3NeMp+27ODV9pOw8pwtK9VtrQ8sk8skqjgu2XTOnQ7GHEd2GlwChGQCgVDHo7AxIBGJMf3Ag==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T13:30:30.872021Z"},"content_sha256":"48418d76c6cc5acc995f2aa11cf330915d2d84106b6f62e6b59a181ccf19c6ac","schema_version":"1.0","event_id":"sha256:48418d76c6cc5acc995f2aa11cf330915d2d84106b6f62e6b59a181ccf19c6ac"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/NJFRIA3Z7CEEHJ4QCG33ACZYPT/bundle.json","state_url":"https://pith.science/pith/NJFRIA3Z7CEEHJ4QCG33ACZYPT/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/NJFRIA3Z7CEEHJ4QCG33ACZYPT/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":"2026-05-31T13:30:30Z","links":{"resolver":"https://pith.science/pith/NJFRIA3Z7CEEHJ4QCG33ACZYPT","bundle":"https://pith.science/pith/NJFRIA3Z7CEEHJ4QCG33ACZYPT/bundle.json","state":"https://pith.science/pith/NJFRIA3Z7CEEHJ4QCG33ACZYPT/state.json","well_known_bundle":"https://pith.science/.well-known/pith/NJFRIA3Z7CEEHJ4QCG33ACZYPT/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:NJFRIA3Z7CEEHJ4QCG33ACZYPT","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":"874a0b0f00ed09f3525436e65a2ea0ad837107bb51956f04931bc81bfba91bee","cross_cats_sorted":["math-ph","math.MP","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-05-05T22:46:14Z","title_canon_sha256":"731434e7fafd7641b9d31a0d1b7a29ad63616debf4cd2c8656fc00c9c39ca86f"},"schema_version":"1.0","source":{"id":"1505.01212","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1505.01212","created_at":"2026-05-18T01:35:38Z"},{"alias_kind":"arxiv_version","alias_value":"1505.01212v3","created_at":"2026-05-18T01:35:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1505.01212","created_at":"2026-05-18T01:35:38Z"},{"alias_kind":"pith_short_12","alias_value":"NJFRIA3Z7CEE","created_at":"2026-05-18T12:29:32Z"},{"alias_kind":"pith_short_16","alias_value":"NJFRIA3Z7CEEHJ4Q","created_at":"2026-05-18T12:29:32Z"},{"alias_kind":"pith_short_8","alias_value":"NJFRIA3Z","created_at":"2026-05-18T12:29:32Z"}],"graph_snapshots":[{"event_id":"sha256:48418d76c6cc5acc995f2aa11cf330915d2d84106b6f62e6b59a181ccf19c6ac","target":"graph","created_at":"2026-05-18T01:35:38Z","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":"In this paper, we study the set of stationary solutions of the Vlasov-Fokker-Planck (VFP) equation. This equation describes the time evolution of the probability distribution of a particle moving under the influence of a double-well potential, an interaction potential, a friction force and a stochastic force. We prove, under suitable assumptions, that the VFP equation does not have a unique stationary solution and that there exists a phase transition. Our study relies on the recent results by Tugaut and coauthors regarding the McKean-Vlasov equation.","authors_text":"Julian Tugaut, Manh Hong Duong","cross_cats":["math-ph","math.MP","math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-05-05T22:46:14Z","title":"Stationary solutions of the Vlasov-Fokker-Planck equation: existence, characterization and phase-transition"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.01212","kind":"arxiv","version":3},"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":"sha256:289dd3f965c25e389cda32b231115b59ba0f42dd513e690a399e9cf2f497f4f6","target":"record","created_at":"2026-05-18T01:35:38Z","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":"874a0b0f00ed09f3525436e65a2ea0ad837107bb51956f04931bc81bfba91bee","cross_cats_sorted":["math-ph","math.MP","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-05-05T22:46:14Z","title_canon_sha256":"731434e7fafd7641b9d31a0d1b7a29ad63616debf4cd2c8656fc00c9c39ca86f"},"schema_version":"1.0","source":{"id":"1505.01212","kind":"arxiv","version":3}},"canonical_sha256":"6a4b140379f88843a79011b7b00b387ce793e1486e31e95b4f12ffde03268d85","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6a4b140379f88843a79011b7b00b387ce793e1486e31e95b4f12ffde03268d85","first_computed_at":"2026-05-18T01:35:38.048836Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:35:38.048836Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"dOFVG38jFsEusRwZ8b+f/OsRScor0ruCsDQohy3maavGj5R5Dsls0gJJkQ6GkOpJgTbzfx+Usn4V1maSfibnAA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:35:38.049456Z","signed_message":"canonical_sha256_bytes"},"source_id":"1505.01212","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:289dd3f965c25e389cda32b231115b59ba0f42dd513e690a399e9cf2f497f4f6","sha256:48418d76c6cc5acc995f2aa11cf330915d2d84106b6f62e6b59a181ccf19c6ac"],"state_sha256":"a1efa24f234f8a775fd93b995e2c6aca4d1f8b22d1b71da17d1e5bce4943b935"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YOQrGWT1v6+60IXhy7KsQau4Zn13zmdFL0rRZt/By2kEb1Ve6plY8nRt6hblpLvAD9HYcIvno0pcUeVPdVWVAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T13:30:30.877743Z","bundle_sha256":"a89ecbae852a23ff5bb44fee7a7e53a9ebba1554cc393abca9c51b49353a0a13"}}