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This form of Onsager's reciprocal relations implies that the shift in time, $\\exp (Kt)$, is also a symmetric operator. This generates the reciprocity relations between the kinetic curves. For example, for the Master equation, if we start the process from the $i$th pure state and measure the probability $p_j(t)$ of the $j$th state ($j\\neq i$), and, similarly, measure $p_i(t)$ for the process, which starts at the $j"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1008.1056","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"cond-mat.stat-mech","submitted_at":"2010-08-05T19:59:29Z","cross_cats_sorted":["physics.chem-ph"],"title_canon_sha256":"a9c5b36829b5301913d477b8942af8aeaf89b6c2b36575d8c06167b3f952b874","abstract_canon_sha256":"fad5c67baf1ce6179e7cb61d6975e82c94e5f3a6890f98884555766963a82d3a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:29:34.825413Z","signature_b64":"UlOEWRdS6jPhZA9ulKepH0TNuA2gEGRDPG0IxInfHvMQsmLe9oZJHaL8xlj08Wh3j/T4+3/aS3TKIVFY79DmCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"58ca426ac9df5d96fdc19a7615c0de7b8f00eccb7564a87df364d339250df791","last_reissued_at":"2026-05-18T04:29:34.824909Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:29:34.824909Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Reciprocal Relations Between Kinetic Curves","license":"http://creativecommons.org/licenses/by/3.0/","headline":"","cross_cats":["physics.chem-ph"],"primary_cat":"cond-mat.stat-mech","authors_text":"A.N. Gorban, D. Constales, G.B. Marin, G.S. Yablonsky, V. Galvita","submitted_at":"2010-08-05T19:59:29Z","abstract_excerpt":"We study coupled irreversible processes. For linear or linearized kinetics with microreversibility, $\\dot{x}=Kx$, the kinetic operator $K$ is symmetric in the entropic inner product. This form of Onsager's reciprocal relations implies that the shift in time, $\\exp (Kt)$, is also a symmetric operator. This generates the reciprocity relations between the kinetic curves. For example, for the Master equation, if we start the process from the $i$th pure state and measure the probability $p_j(t)$ of the $j$th state ($j\\neq i$), and, similarly, measure $p_i(t)$ for the process, which starts at the $j"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1008.1056","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1008.1056","created_at":"2026-05-18T04:29:34.824984+00:00"},{"alias_kind":"arxiv_version","alias_value":"1008.1056v2","created_at":"2026-05-18T04:29:34.824984+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1008.1056","created_at":"2026-05-18T04:29:34.824984+00:00"},{"alias_kind":"pith_short_12","alias_value":"LDFEE2WJ35OZ","created_at":"2026-05-18T12:26:10.704358+00:00"},{"alias_kind":"pith_short_16","alias_value":"LDFEE2WJ35OZN7OB","created_at":"2026-05-18T12:26:10.704358+00:00"},{"alias_kind":"pith_short_8","alias_value":"LDFEE2WJ","created_at":"2026-05-18T12:26:10.704358+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/LDFEE2WJ35OZN7OBTJ3BLQG6PO","json":"https://pith.science/pith/LDFEE2WJ35OZN7OBTJ3BLQG6PO.json","graph_json":"https://pith.science/api/pith-number/LDFEE2WJ35OZN7OBTJ3BLQG6PO/graph.json","events_json":"https://pith.science/api/pith-number/LDFEE2WJ35OZN7OBTJ3BLQG6PO/events.json","paper":"https://pith.science/paper/LDFEE2WJ"},"agent_actions":{"view_html":"https://pith.science/pith/LDFEE2WJ35OZN7OBTJ3BLQG6PO","download_json":"https://pith.science/pith/LDFEE2WJ35OZN7OBTJ3BLQG6PO.json","view_paper":"https://pith.science/paper/LDFEE2WJ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1008.1056&json=true","fetch_graph":"https://pith.science/api/pith-number/LDFEE2WJ35OZN7OBTJ3BLQG6PO/graph.json","fetch_events":"https://pith.science/api/pith-number/LDFEE2WJ35OZN7OBTJ3BLQG6PO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/LDFEE2WJ35OZN7OBTJ3BLQG6PO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/LDFEE2WJ35OZN7OBTJ3BLQG6PO/action/storage_attestation","attest_author":"https://pith.science/pith/LDFEE2WJ35OZN7OBTJ3BLQG6PO/action/author_attestation","sign_citation":"https://pith.science/pith/LDFEE2WJ35OZN7OBTJ3BLQG6PO/action/citation_signature","submit_replication":"https://pith.science/pith/LDFEE2WJ35OZN7OBTJ3BLQG6PO/action/replication_record"}},"created_at":"2026-05-18T04:29:34.824984+00:00","updated_at":"2026-05-18T04:29:34.824984+00:00"}