{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:ETL4KXKSB3ARFTMHSTYHQL43PT","short_pith_number":"pith:ETL4KXKS","canonical_record":{"source":{"id":"1302.2756","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2013-02-12T10:56:16Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"b0781650900372fd19cc53aafdad0c45c3fe06e74e87840fe205f23a993ecd98","abstract_canon_sha256":"9c4dbd86e6632d7b67db8d5cc9f500cbc490d5c97d68be40e3530a9966860b41"},"schema_version":"1.0"},"canonical_sha256":"24d7c55d520ec112cd8794f0782f9b7cc2500fd6e6fe7dd4083b6d5eccb7ea7a","source":{"kind":"arxiv","id":"1302.2756","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1302.2756","created_at":"2026-05-18T00:34:00Z"},{"alias_kind":"arxiv_version","alias_value":"1302.2756v2","created_at":"2026-05-18T00:34:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1302.2756","created_at":"2026-05-18T00:34:00Z"},{"alias_kind":"pith_short_12","alias_value":"ETL4KXKSB3AR","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_16","alias_value":"ETL4KXKSB3ARFTMH","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_8","alias_value":"ETL4KXKS","created_at":"2026-05-18T12:27:43Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:ETL4KXKSB3ARFTMHSTYHQL43PT","target":"record","payload":{"canonical_record":{"source":{"id":"1302.2756","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2013-02-12T10:56:16Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"b0781650900372fd19cc53aafdad0c45c3fe06e74e87840fe205f23a993ecd98","abstract_canon_sha256":"9c4dbd86e6632d7b67db8d5cc9f500cbc490d5c97d68be40e3530a9966860b41"},"schema_version":"1.0"},"canonical_sha256":"24d7c55d520ec112cd8794f0782f9b7cc2500fd6e6fe7dd4083b6d5eccb7ea7a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:34:00.678324Z","signature_b64":"tHaqB2CMLr4SzTb44MHJZO0nLCzLD6GEXtoz4SZkJvW6kxREbHVFWiuAT8DT4L0gsdoCbEzlyjiijgr93+hKBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"24d7c55d520ec112cd8794f0782f9b7cc2500fd6e6fe7dd4083b6d5eccb7ea7a","last_reissued_at":"2026-05-18T00:34:00.677625Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:34:00.677625Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1302.2756","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":"2026-05-18T00:34:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LxB4N0RpzqjJiWcqLWvq6fw8SQrUOv92OBT6GpoGdqCF9ylpJnilhi+v7PW9PkVpuwQ6f5nczgKXAaVXLgEsAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T20:10:55.856563Z"},"content_sha256":"acf259527b857b643f6be539336d1b7b0928ce33b4b49221f927ca8709b4d43f","schema_version":"1.0","event_id":"sha256:acf259527b857b643f6be539336d1b7b0928ce33b4b49221f927ca8709b4d43f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:ETL4KXKSB3ARFTMHSTYHQL43PT","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the convergence to a statistical equilibrium for the wave equations coupled to a particle","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"T.V. Dudnikova","submitted_at":"2013-02-12T10:56:16Z","abstract_excerpt":"We consider a linear Hamiltonian system consisting of a classical particle and a scalar field describing by the wave or Klein-Gordon equations with variable coefficients. The initial data of the system are supposed to be a random function which has some mixing properties. We study the distribution \\mu_t of the random solution at time moments t\\in\\R. The main result is the convergence of \\mu_t to a Gaussian probability measure as t\\to\\infty. The mixing properties of the limit measures are studied. The application to the case of Gibbs initial measures is given."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1302.2756","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":"2026-05-18T00:34:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"x3Pyiyfq8hxBG6AzgDg9c1OdiGPy//8YGnItFkGl5yMhkO38mf+BKgNJXh/Kgu0HYVnLz0IjJH/CWjencL8XAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T20:10:55.856918Z"},"content_sha256":"2191ddb38514570e232bb4d84a29d71e2ceb5a60396fa1c6949a6da7fc305c10","schema_version":"1.0","event_id":"sha256:2191ddb38514570e232bb4d84a29d71e2ceb5a60396fa1c6949a6da7fc305c10"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ETL4KXKSB3ARFTMHSTYHQL43PT/bundle.json","state_url":"https://pith.science/pith/ETL4KXKSB3ARFTMHSTYHQL43PT/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ETL4KXKSB3ARFTMHSTYHQL43PT/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":"2026-06-26T20:10:55Z","links":{"resolver":"https://pith.science/pith/ETL4KXKSB3ARFTMHSTYHQL43PT","bundle":"https://pith.science/pith/ETL4KXKSB3ARFTMHSTYHQL43PT/bundle.json","state":"https://pith.science/pith/ETL4KXKSB3ARFTMHSTYHQL43PT/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ETL4KXKSB3ARFTMHSTYHQL43PT/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:ETL4KXKSB3ARFTMHSTYHQL43PT","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":"9c4dbd86e6632d7b67db8d5cc9f500cbc490d5c97d68be40e3530a9966860b41","cross_cats_sorted":["math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2013-02-12T10:56:16Z","title_canon_sha256":"b0781650900372fd19cc53aafdad0c45c3fe06e74e87840fe205f23a993ecd98"},"schema_version":"1.0","source":{"id":"1302.2756","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1302.2756","created_at":"2026-05-18T00:34:00Z"},{"alias_kind":"arxiv_version","alias_value":"1302.2756v2","created_at":"2026-05-18T00:34:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1302.2756","created_at":"2026-05-18T00:34:00Z"},{"alias_kind":"pith_short_12","alias_value":"ETL4KXKSB3AR","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_16","alias_value":"ETL4KXKSB3ARFTMH","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_8","alias_value":"ETL4KXKS","created_at":"2026-05-18T12:27:43Z"}],"graph_snapshots":[{"event_id":"sha256:2191ddb38514570e232bb4d84a29d71e2ceb5a60396fa1c6949a6da7fc305c10","target":"graph","created_at":"2026-05-18T00:34:00Z","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 consider a linear Hamiltonian system consisting of a classical particle and a scalar field describing by the wave or Klein-Gordon equations with variable coefficients. The initial data of the system are supposed to be a random function which has some mixing properties. We study the distribution \\mu_t of the random solution at time moments t\\in\\R. The main result is the convergence of \\mu_t to a Gaussian probability measure as t\\to\\infty. The mixing properties of the limit measures are studied. The application to the case of Gibbs initial measures is given.","authors_text":"T.V. Dudnikova","cross_cats":["math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2013-02-12T10:56:16Z","title":"On the convergence to a statistical equilibrium for the wave equations coupled to a particle"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1302.2756","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:acf259527b857b643f6be539336d1b7b0928ce33b4b49221f927ca8709b4d43f","target":"record","created_at":"2026-05-18T00:34:00Z","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":"9c4dbd86e6632d7b67db8d5cc9f500cbc490d5c97d68be40e3530a9966860b41","cross_cats_sorted":["math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2013-02-12T10:56:16Z","title_canon_sha256":"b0781650900372fd19cc53aafdad0c45c3fe06e74e87840fe205f23a993ecd98"},"schema_version":"1.0","source":{"id":"1302.2756","kind":"arxiv","version":2}},"canonical_sha256":"24d7c55d520ec112cd8794f0782f9b7cc2500fd6e6fe7dd4083b6d5eccb7ea7a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"24d7c55d520ec112cd8794f0782f9b7cc2500fd6e6fe7dd4083b6d5eccb7ea7a","first_computed_at":"2026-05-18T00:34:00.677625Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:34:00.677625Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"tHaqB2CMLr4SzTb44MHJZO0nLCzLD6GEXtoz4SZkJvW6kxREbHVFWiuAT8DT4L0gsdoCbEzlyjiijgr93+hKBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:34:00.678324Z","signed_message":"canonical_sha256_bytes"},"source_id":"1302.2756","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:acf259527b857b643f6be539336d1b7b0928ce33b4b49221f927ca8709b4d43f","sha256:2191ddb38514570e232bb4d84a29d71e2ceb5a60396fa1c6949a6da7fc305c10"],"state_sha256":"85fd1dc09e491f08898e1e34c44632bf2fd76af39c92d9d39a61a61d23fadaeb"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cQZ064nPVDDkSn/+OdbEnvxc2wsf4MBAZByerpAIZNHGqaCQH5NEevEFkYGZJATZJPUpXfqzPEyPbAX4Ny+ZCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-26T20:10:55.858895Z","bundle_sha256":"e504a08fad49bd3676ec83d59dc2c339b8200dc4011b07038a8a914c22e65e74"}}