{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:Z4UKD23QHMSX7J6ARK4AVQJZKI","short_pith_number":"pith:Z4UKD23Q","canonical_record":{"source":{"id":"1504.08317","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"physics.chem-ph","submitted_at":"2015-04-30T17:49:28Z","cross_cats_sorted":["cond-mat.stat-mech"],"title_canon_sha256":"2d64211b56f8cf34e524096f24ccf37b1693aab94fb7fe25805bfa5e1a3996ba","abstract_canon_sha256":"c79848dc0f496ec7d04ba716a95dcd9723ec90a2d98125778f9dbe655617a8d8"},"schema_version":"1.0"},"canonical_sha256":"cf28a1eb703b257fa7c08ab80ac1395223ee0a55507b181b9f0777399d905095","source":{"kind":"arxiv","id":"1504.08317","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1504.08317","created_at":"2026-05-18T01:32:08Z"},{"alias_kind":"arxiv_version","alias_value":"1504.08317v1","created_at":"2026-05-18T01:32:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.08317","created_at":"2026-05-18T01:32:08Z"},{"alias_kind":"pith_short_12","alias_value":"Z4UKD23QHMSX","created_at":"2026-05-18T12:29:52Z"},{"alias_kind":"pith_short_16","alias_value":"Z4UKD23QHMSX7J6A","created_at":"2026-05-18T12:29:52Z"},{"alias_kind":"pith_short_8","alias_value":"Z4UKD23Q","created_at":"2026-05-18T12:29:52Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:Z4UKD23QHMSX7J6ARK4AVQJZKI","target":"record","payload":{"canonical_record":{"source":{"id":"1504.08317","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"physics.chem-ph","submitted_at":"2015-04-30T17:49:28Z","cross_cats_sorted":["cond-mat.stat-mech"],"title_canon_sha256":"2d64211b56f8cf34e524096f24ccf37b1693aab94fb7fe25805bfa5e1a3996ba","abstract_canon_sha256":"c79848dc0f496ec7d04ba716a95dcd9723ec90a2d98125778f9dbe655617a8d8"},"schema_version":"1.0"},"canonical_sha256":"cf28a1eb703b257fa7c08ab80ac1395223ee0a55507b181b9f0777399d905095","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:32:08.038825Z","signature_b64":"yK95KrZMCy9TYWF0pSeKsTM76OjmhPY3AEdASLwvxm2gPgLc78j6koCWTYafb0TowtMbMHAvQwn8LKzAEFoFBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cf28a1eb703b257fa7c08ab80ac1395223ee0a55507b181b9f0777399d905095","last_reissued_at":"2026-05-18T01:32:08.038263Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:32:08.038263Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1504.08317","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":"2026-05-18T01:32:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"H3N5PLXImZai9q6Yjo3GR6z3DQ+JQ8OtiLh8jx9yQ0mWUoujn+tYoyDmxSjVS9lS1ivPuVHq+BfAXve3TrSaAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T14:37:42.650590Z"},"content_sha256":"e8cebeb8f75941524de052ff02476632b0a8a5b238aa2ed35037c8c000ee4add","schema_version":"1.0","event_id":"sha256:e8cebeb8f75941524de052ff02476632b0a8a5b238aa2ed35037c8c000ee4add"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:Z4UKD23QHMSX7J6ARK4AVQJZKI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Generalized Mass Action Law and Thermodynamics of Nonlinear Markov Processes","license":"http://creativecommons.org/licenses/by/3.0/","headline":"","cross_cats":["cond-mat.stat-mech"],"primary_cat":"physics.chem-ph","authors_text":"A.N. Gorban, V.N. Kolokoltsov","submitted_at":"2015-04-30T17:49:28Z","abstract_excerpt":"The nonlinear Markov processes are the measure-valued dynamical systems which preserve positivity. They can be represented as the law of large numbers limits of general Markov models of interacting particles. In physics, the kinetic equations allow Lyapunov functionals (entropy, free energy, etc.). This may be considered as a sort of inheritance of the Lyapunov functionals from the microscopic master equations. We study nonlinear Markov processes that inherit thermodynamic properties from the microscopic linear Markov processes. We develop the thermodynamics of nonlinear Markov processes and a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.08317","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":"2026-05-18T01:32:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2qu5Nie+FSgtDOOlkaP3JizxgTk1S445M7FZPD0OQuf82RfM2yD0TCpwucfTtqyrkPdGKw95LnZYKe1cr9/CAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T14:37:42.650963Z"},"content_sha256":"647fecdf5fd1fa1af26e45f7a2d589fc5816fd2dbe3529b3a7261244b1f31cb4","schema_version":"1.0","event_id":"sha256:647fecdf5fd1fa1af26e45f7a2d589fc5816fd2dbe3529b3a7261244b1f31cb4"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/Z4UKD23QHMSX7J6ARK4AVQJZKI/bundle.json","state_url":"https://pith.science/pith/Z4UKD23QHMSX7J6ARK4AVQJZKI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/Z4UKD23QHMSX7J6ARK4AVQJZKI/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":"2026-05-28T14:37:42Z","links":{"resolver":"https://pith.science/pith/Z4UKD23QHMSX7J6ARK4AVQJZKI","bundle":"https://pith.science/pith/Z4UKD23QHMSX7J6ARK4AVQJZKI/bundle.json","state":"https://pith.science/pith/Z4UKD23QHMSX7J6ARK4AVQJZKI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/Z4UKD23QHMSX7J6ARK4AVQJZKI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:Z4UKD23QHMSX7J6ARK4AVQJZKI","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":"c79848dc0f496ec7d04ba716a95dcd9723ec90a2d98125778f9dbe655617a8d8","cross_cats_sorted":["cond-mat.stat-mech"],"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"physics.chem-ph","submitted_at":"2015-04-30T17:49:28Z","title_canon_sha256":"2d64211b56f8cf34e524096f24ccf37b1693aab94fb7fe25805bfa5e1a3996ba"},"schema_version":"1.0","source":{"id":"1504.08317","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1504.08317","created_at":"2026-05-18T01:32:08Z"},{"alias_kind":"arxiv_version","alias_value":"1504.08317v1","created_at":"2026-05-18T01:32:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.08317","created_at":"2026-05-18T01:32:08Z"},{"alias_kind":"pith_short_12","alias_value":"Z4UKD23QHMSX","created_at":"2026-05-18T12:29:52Z"},{"alias_kind":"pith_short_16","alias_value":"Z4UKD23QHMSX7J6A","created_at":"2026-05-18T12:29:52Z"},{"alias_kind":"pith_short_8","alias_value":"Z4UKD23Q","created_at":"2026-05-18T12:29:52Z"}],"graph_snapshots":[{"event_id":"sha256:647fecdf5fd1fa1af26e45f7a2d589fc5816fd2dbe3529b3a7261244b1f31cb4","target":"graph","created_at":"2026-05-18T01:32:08Z","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 nonlinear Markov processes are the measure-valued dynamical systems which preserve positivity. They can be represented as the law of large numbers limits of general Markov models of interacting particles. In physics, the kinetic equations allow Lyapunov functionals (entropy, free energy, etc.). This may be considered as a sort of inheritance of the Lyapunov functionals from the microscopic master equations. We study nonlinear Markov processes that inherit thermodynamic properties from the microscopic linear Markov processes. We develop the thermodynamics of nonlinear Markov processes and a","authors_text":"A.N. Gorban, V.N. Kolokoltsov","cross_cats":["cond-mat.stat-mech"],"headline":"","license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"physics.chem-ph","submitted_at":"2015-04-30T17:49:28Z","title":"Generalized Mass Action Law and Thermodynamics of Nonlinear Markov Processes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.08317","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:e8cebeb8f75941524de052ff02476632b0a8a5b238aa2ed35037c8c000ee4add","target":"record","created_at":"2026-05-18T01:32:08Z","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":"c79848dc0f496ec7d04ba716a95dcd9723ec90a2d98125778f9dbe655617a8d8","cross_cats_sorted":["cond-mat.stat-mech"],"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"physics.chem-ph","submitted_at":"2015-04-30T17:49:28Z","title_canon_sha256":"2d64211b56f8cf34e524096f24ccf37b1693aab94fb7fe25805bfa5e1a3996ba"},"schema_version":"1.0","source":{"id":"1504.08317","kind":"arxiv","version":1}},"canonical_sha256":"cf28a1eb703b257fa7c08ab80ac1395223ee0a55507b181b9f0777399d905095","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"cf28a1eb703b257fa7c08ab80ac1395223ee0a55507b181b9f0777399d905095","first_computed_at":"2026-05-18T01:32:08.038263Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:32:08.038263Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"yK95KrZMCy9TYWF0pSeKsTM76OjmhPY3AEdASLwvxm2gPgLc78j6koCWTYafb0TowtMbMHAvQwn8LKzAEFoFBw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:32:08.038825Z","signed_message":"canonical_sha256_bytes"},"source_id":"1504.08317","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e8cebeb8f75941524de052ff02476632b0a8a5b238aa2ed35037c8c000ee4add","sha256:647fecdf5fd1fa1af26e45f7a2d589fc5816fd2dbe3529b3a7261244b1f31cb4"],"state_sha256":"38986ead533a8196212692681d9b24e2afb3e357475e51896c7a8e70bad7f1fb"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HT7gxSyyfWq87/4g1SuZcJknZcePrREEbhpOFbsPDynSuFykFLSc7YXUOW5tjyIX36BNSMoBR7YK8MvIQmnnCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T14:37:42.653070Z","bundle_sha256":"550ab309ba4e8de5a739c23db50ee9d2ffc542269661470e98634a01b83ad67a"}}