{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:NJVNRKHVNPM33LY4JWZZFK2ZID","short_pith_number":"pith:NJVNRKHV","canonical_record":{"source":{"id":"1212.3920","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2012-12-17T08:14:52Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"aa03de270ce21486890e586baeed11ddcf1d59dfeb5c120e2d96b289b2b71366","abstract_canon_sha256":"0b34afb45803f6cca971a8162a72795dde7da9e262673fd37ea7cfa178186a33"},"schema_version":"1.0"},"canonical_sha256":"6a6ad8a8f56bd9bdaf1c4db392ab5940dde662c09f6a5695ea68d4bdb8388999","source":{"kind":"arxiv","id":"1212.3920","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1212.3920","created_at":"2026-05-18T03:38:22Z"},{"alias_kind":"arxiv_version","alias_value":"1212.3920v1","created_at":"2026-05-18T03:38:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1212.3920","created_at":"2026-05-18T03:38:22Z"},{"alias_kind":"pith_short_12","alias_value":"NJVNRKHVNPM3","created_at":"2026-05-18T12:27:16Z"},{"alias_kind":"pith_short_16","alias_value":"NJVNRKHVNPM33LY4","created_at":"2026-05-18T12:27:16Z"},{"alias_kind":"pith_short_8","alias_value":"NJVNRKHV","created_at":"2026-05-18T12:27:16Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:NJVNRKHVNPM33LY4JWZZFK2ZID","target":"record","payload":{"canonical_record":{"source":{"id":"1212.3920","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2012-12-17T08:14:52Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"aa03de270ce21486890e586baeed11ddcf1d59dfeb5c120e2d96b289b2b71366","abstract_canon_sha256":"0b34afb45803f6cca971a8162a72795dde7da9e262673fd37ea7cfa178186a33"},"schema_version":"1.0"},"canonical_sha256":"6a6ad8a8f56bd9bdaf1c4db392ab5940dde662c09f6a5695ea68d4bdb8388999","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:38:22.057629Z","signature_b64":"l9Hsr49hIqxW/0VXecTTIKzm8wmpDvpuT4JHpe22+CBHN3qcNUhqdjHG4NTgPmPTflH1gofOP6FaqGj/lwEbCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6a6ad8a8f56bd9bdaf1c4db392ab5940dde662c09f6a5695ea68d4bdb8388999","last_reissued_at":"2026-05-18T03:38:22.057110Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:38:22.057110Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1212.3920","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":"2026-05-18T03:38:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wovkKGMAfX/6SBuS8fCDohPgZGFZV1vBFdfBMsLBU3Y5+ng2YYUlEIiPqAQi4SwHfDIwhvuTk5jFidSMZfUPBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T22:48:27.067138Z"},"content_sha256":"d20246db912268f5cac393cc741ea000f46954a930fb643cdc352493fc951dd7","schema_version":"1.0","event_id":"sha256:d20246db912268f5cac393cc741ea000f46954a930fb643cdc352493fc951dd7"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:NJVNRKHVNPM33LY4JWZZFK2ZID","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A note on phase transitions for the Smoluchowski equation with dipolar potential","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Amic Frouvelle (CEREMADE), Jian-Guo Liu, Pierre Degond (IMT)","submitted_at":"2012-12-17T08:14:52Z","abstract_excerpt":"In this note, we study the phase transitions arising in a modified Smoluchowski equation on the sphere with dipolar potential. This equation models the competition between alignment and diffusion, and the modification consists in taking the strength of alignment and the intensity of the diffusion as functions of the order parameter. We characterize the stable and unstable equilibrium states. For stable equilibria, we provide the exponential rate of convergence. We detail special cases, giving rise to second order and first order phase transitions, respectively. We study the hysteresis diagram,"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.3920","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":"2026-05-18T03:38:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"X9QJnIbtfuek9dUuIKqM6CQQyJ6JlLgYpZ9yMDY61PsOgw7Siwpv2N3HAblzp8OOEnhF8X5Bfx1uRgs6Or2ECg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T22:48:27.067510Z"},"content_sha256":"b26c567dd416bb14f16f0ec295b7d3f26f6d253ec681a852f928f773fab57931","schema_version":"1.0","event_id":"sha256:b26c567dd416bb14f16f0ec295b7d3f26f6d253ec681a852f928f773fab57931"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/NJVNRKHVNPM33LY4JWZZFK2ZID/bundle.json","state_url":"https://pith.science/pith/NJVNRKHVNPM33LY4JWZZFK2ZID/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/NJVNRKHVNPM33LY4JWZZFK2ZID/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":"2026-06-23T22:48:27Z","links":{"resolver":"https://pith.science/pith/NJVNRKHVNPM33LY4JWZZFK2ZID","bundle":"https://pith.science/pith/NJVNRKHVNPM33LY4JWZZFK2ZID/bundle.json","state":"https://pith.science/pith/NJVNRKHVNPM33LY4JWZZFK2ZID/state.json","well_known_bundle":"https://pith.science/.well-known/pith/NJVNRKHVNPM33LY4JWZZFK2ZID/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:NJVNRKHVNPM33LY4JWZZFK2ZID","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":"0b34afb45803f6cca971a8162a72795dde7da9e262673fd37ea7cfa178186a33","cross_cats_sorted":["math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2012-12-17T08:14:52Z","title_canon_sha256":"aa03de270ce21486890e586baeed11ddcf1d59dfeb5c120e2d96b289b2b71366"},"schema_version":"1.0","source":{"id":"1212.3920","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1212.3920","created_at":"2026-05-18T03:38:22Z"},{"alias_kind":"arxiv_version","alias_value":"1212.3920v1","created_at":"2026-05-18T03:38:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1212.3920","created_at":"2026-05-18T03:38:22Z"},{"alias_kind":"pith_short_12","alias_value":"NJVNRKHVNPM3","created_at":"2026-05-18T12:27:16Z"},{"alias_kind":"pith_short_16","alias_value":"NJVNRKHVNPM33LY4","created_at":"2026-05-18T12:27:16Z"},{"alias_kind":"pith_short_8","alias_value":"NJVNRKHV","created_at":"2026-05-18T12:27:16Z"}],"graph_snapshots":[{"event_id":"sha256:b26c567dd416bb14f16f0ec295b7d3f26f6d253ec681a852f928f773fab57931","target":"graph","created_at":"2026-05-18T03:38:22Z","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 note, we study the phase transitions arising in a modified Smoluchowski equation on the sphere with dipolar potential. This equation models the competition between alignment and diffusion, and the modification consists in taking the strength of alignment and the intensity of the diffusion as functions of the order parameter. We characterize the stable and unstable equilibrium states. For stable equilibria, we provide the exponential rate of convergence. We detail special cases, giving rise to second order and first order phase transitions, respectively. We study the hysteresis diagram,","authors_text":"Amic Frouvelle (CEREMADE), Jian-Guo Liu, Pierre Degond (IMT)","cross_cats":["math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2012-12-17T08:14:52Z","title":"A note on phase transitions for the Smoluchowski equation with dipolar potential"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.3920","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:d20246db912268f5cac393cc741ea000f46954a930fb643cdc352493fc951dd7","target":"record","created_at":"2026-05-18T03:38:22Z","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":"0b34afb45803f6cca971a8162a72795dde7da9e262673fd37ea7cfa178186a33","cross_cats_sorted":["math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2012-12-17T08:14:52Z","title_canon_sha256":"aa03de270ce21486890e586baeed11ddcf1d59dfeb5c120e2d96b289b2b71366"},"schema_version":"1.0","source":{"id":"1212.3920","kind":"arxiv","version":1}},"canonical_sha256":"6a6ad8a8f56bd9bdaf1c4db392ab5940dde662c09f6a5695ea68d4bdb8388999","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6a6ad8a8f56bd9bdaf1c4db392ab5940dde662c09f6a5695ea68d4bdb8388999","first_computed_at":"2026-05-18T03:38:22.057110Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:38:22.057110Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"l9Hsr49hIqxW/0VXecTTIKzm8wmpDvpuT4JHpe22+CBHN3qcNUhqdjHG4NTgPmPTflH1gofOP6FaqGj/lwEbCA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:38:22.057629Z","signed_message":"canonical_sha256_bytes"},"source_id":"1212.3920","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d20246db912268f5cac393cc741ea000f46954a930fb643cdc352493fc951dd7","sha256:b26c567dd416bb14f16f0ec295b7d3f26f6d253ec681a852f928f773fab57931"],"state_sha256":"68452e144aee769efb6a23cfe1f1719e11a45d3a896687bd8b54f025ed6669d0"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rSmn+Sh6MnWWgzgi+MiM1Bta1r8M3GnbQVGSAc8ZQAgJDdCGCo6Af5UHIZ02TNhf9NkhMEE0k1/xHXlxxOGEBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T22:48:27.069420Z","bundle_sha256":"c5e3b72d23c8b229695f6323b0f6c698e93d70526e6426d825e924d39f70e23c"}}