{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:1996:NKRT2FISMMC5YVMXEKYUJXJAO3","short_pith_number":"pith:NKRT2FIS","canonical_record":{"source":{"id":"chao-dyn/9607019","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"chao-dyn","submitted_at":"1996-07-31T16:19:42Z","cross_cats_sorted":["nlin.CD"],"title_canon_sha256":"c04eca8f8d60648b124b172048026add19d7115baa98f4b8f62b8549c08ff4be","abstract_canon_sha256":"eb4832a04df334a057019295059a8289b6af793689dc15b2c0a7a706b1d3df8c"},"schema_version":"1.0"},"canonical_sha256":"6aa33d15126305dc559722b144dd2076f40d722f92c20976eecd18e2af9a5e73","source":{"kind":"arxiv","id":"chao-dyn/9607019","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"chao-dyn/9607019","created_at":"2026-05-18T00:34:45Z"},{"alias_kind":"arxiv_version","alias_value":"chao-dyn/9607019v1","created_at":"2026-05-18T00:34:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.chao-dyn/9607019","created_at":"2026-05-18T00:34:45Z"},{"alias_kind":"pith_short_12","alias_value":"NKRT2FISMMC5","created_at":"2026-05-18T12:25:48Z"},{"alias_kind":"pith_short_16","alias_value":"NKRT2FISMMC5YVMX","created_at":"2026-05-18T12:25:48Z"},{"alias_kind":"pith_short_8","alias_value":"NKRT2FIS","created_at":"2026-05-18T12:25:48Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:1996:NKRT2FISMMC5YVMXEKYUJXJAO3","target":"record","payload":{"canonical_record":{"source":{"id":"chao-dyn/9607019","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"chao-dyn","submitted_at":"1996-07-31T16:19:42Z","cross_cats_sorted":["nlin.CD"],"title_canon_sha256":"c04eca8f8d60648b124b172048026add19d7115baa98f4b8f62b8549c08ff4be","abstract_canon_sha256":"eb4832a04df334a057019295059a8289b6af793689dc15b2c0a7a706b1d3df8c"},"schema_version":"1.0"},"canonical_sha256":"6aa33d15126305dc559722b144dd2076f40d722f92c20976eecd18e2af9a5e73","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:34:45.392875Z","signature_b64":"zYftmJhJ7Nlniv+5005wACImUfG37dBONARdK51lEou3zVZ2eot0Z1450HgPuaHg+i2nrfKfRsM5vZmH5+v4CA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6aa33d15126305dc559722b144dd2076f40d722f92c20976eecd18e2af9a5e73","last_reissued_at":"2026-05-18T00:34:45.392138Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:34:45.392138Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"chao-dyn/9607019","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":"2026-05-18T00:34:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PKSVByXLwjOa0aL1ZXJEeMIKoPZXcuyO3cw/8aKIPKcd+Ifzl6AuC5+IgfagB1K+Gf4p55+IgxLAJ7LQqrCWCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T01:22:44.275620Z"},"content_sha256":"be23ea695e25956ca55f79fc47caf241ec0706eb029342fa8fcd8eb8310d6f74","schema_version":"1.0","event_id":"sha256:be23ea695e25956ca55f79fc47caf241ec0706eb029342fa8fcd8eb8310d6f74"},{"event_type":"graph_snapshot","subject_pith_number":"pith:1996:NKRT2FISMMC5YVMXEKYUJXJAO3","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Thermodynamic formalism and localization in Lorentz gases and hopping models","license":"","headline":"","cross_cats":["nlin.CD"],"primary_cat":"chao-dyn","authors_text":"C. Appert, H. van Beijeren, J.R. Dorfman, M.H. Ernst","submitted_at":"1996-07-31T16:19:42Z","abstract_excerpt":"The thermodynamic formalism expresses chaotic properties of dynamical systems in terms of the Ruelle pressure $\\psi(\\beta)$. The inverse-temperature like variable $\\beta$ allows one to scan the structure of the probability distribution in the dynamic phase space. This formalism is applied here to a Lorentz Lattice Gas, where a particle moving on a lattice of size $L^d$ collides with fixed scatterers placed at random locations. Here we give rigorous arguments that the Ruelle pressure in the limit of infinit e systems has two branches joining with a slope discontinuity at $\\beta = 1$. The low an"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"chao-dyn/9607019","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":"2026-05-18T00:34:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"X5JRSXizjNlRS14BrNVadQZ7qUDW3zJExLZT7M7086EaPO9u3XFrXjDCTLSFOhUchojRJVnaUCbQ64ka7nXEBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T01:22:44.275974Z"},"content_sha256":"8ef3a2d735a09c3717337848bbeb5b75d4a79f033041a4df1b129834464eddd3","schema_version":"1.0","event_id":"sha256:8ef3a2d735a09c3717337848bbeb5b75d4a79f033041a4df1b129834464eddd3"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/NKRT2FISMMC5YVMXEKYUJXJAO3/bundle.json","state_url":"https://pith.science/pith/NKRT2FISMMC5YVMXEKYUJXJAO3/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/NKRT2FISMMC5YVMXEKYUJXJAO3/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":"2026-06-02T01:22:44Z","links":{"resolver":"https://pith.science/pith/NKRT2FISMMC5YVMXEKYUJXJAO3","bundle":"https://pith.science/pith/NKRT2FISMMC5YVMXEKYUJXJAO3/bundle.json","state":"https://pith.science/pith/NKRT2FISMMC5YVMXEKYUJXJAO3/state.json","well_known_bundle":"https://pith.science/.well-known/pith/NKRT2FISMMC5YVMXEKYUJXJAO3/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:1996:NKRT2FISMMC5YVMXEKYUJXJAO3","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":"eb4832a04df334a057019295059a8289b6af793689dc15b2c0a7a706b1d3df8c","cross_cats_sorted":["nlin.CD"],"license":"","primary_cat":"chao-dyn","submitted_at":"1996-07-31T16:19:42Z","title_canon_sha256":"c04eca8f8d60648b124b172048026add19d7115baa98f4b8f62b8549c08ff4be"},"schema_version":"1.0","source":{"id":"chao-dyn/9607019","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"chao-dyn/9607019","created_at":"2026-05-18T00:34:45Z"},{"alias_kind":"arxiv_version","alias_value":"chao-dyn/9607019v1","created_at":"2026-05-18T00:34:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.chao-dyn/9607019","created_at":"2026-05-18T00:34:45Z"},{"alias_kind":"pith_short_12","alias_value":"NKRT2FISMMC5","created_at":"2026-05-18T12:25:48Z"},{"alias_kind":"pith_short_16","alias_value":"NKRT2FISMMC5YVMX","created_at":"2026-05-18T12:25:48Z"},{"alias_kind":"pith_short_8","alias_value":"NKRT2FIS","created_at":"2026-05-18T12:25:48Z"}],"graph_snapshots":[{"event_id":"sha256:8ef3a2d735a09c3717337848bbeb5b75d4a79f033041a4df1b129834464eddd3","target":"graph","created_at":"2026-05-18T00:34:45Z","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 thermodynamic formalism expresses chaotic properties of dynamical systems in terms of the Ruelle pressure $\\psi(\\beta)$. The inverse-temperature like variable $\\beta$ allows one to scan the structure of the probability distribution in the dynamic phase space. This formalism is applied here to a Lorentz Lattice Gas, where a particle moving on a lattice of size $L^d$ collides with fixed scatterers placed at random locations. Here we give rigorous arguments that the Ruelle pressure in the limit of infinit e systems has two branches joining with a slope discontinuity at $\\beta = 1$. The low an","authors_text":"C. Appert, H. van Beijeren, J.R. Dorfman, M.H. Ernst","cross_cats":["nlin.CD"],"headline":"","license":"","primary_cat":"chao-dyn","submitted_at":"1996-07-31T16:19:42Z","title":"Thermodynamic formalism and localization in Lorentz gases and hopping models"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"chao-dyn/9607019","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:be23ea695e25956ca55f79fc47caf241ec0706eb029342fa8fcd8eb8310d6f74","target":"record","created_at":"2026-05-18T00:34:45Z","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":"eb4832a04df334a057019295059a8289b6af793689dc15b2c0a7a706b1d3df8c","cross_cats_sorted":["nlin.CD"],"license":"","primary_cat":"chao-dyn","submitted_at":"1996-07-31T16:19:42Z","title_canon_sha256":"c04eca8f8d60648b124b172048026add19d7115baa98f4b8f62b8549c08ff4be"},"schema_version":"1.0","source":{"id":"chao-dyn/9607019","kind":"arxiv","version":1}},"canonical_sha256":"6aa33d15126305dc559722b144dd2076f40d722f92c20976eecd18e2af9a5e73","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6aa33d15126305dc559722b144dd2076f40d722f92c20976eecd18e2af9a5e73","first_computed_at":"2026-05-18T00:34:45.392138Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:34:45.392138Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"zYftmJhJ7Nlniv+5005wACImUfG37dBONARdK51lEou3zVZ2eot0Z1450HgPuaHg+i2nrfKfRsM5vZmH5+v4CA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:34:45.392875Z","signed_message":"canonical_sha256_bytes"},"source_id":"chao-dyn/9607019","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:be23ea695e25956ca55f79fc47caf241ec0706eb029342fa8fcd8eb8310d6f74","sha256:8ef3a2d735a09c3717337848bbeb5b75d4a79f033041a4df1b129834464eddd3"],"state_sha256":"980c9129dc85752e57e62aad0fea92a41dde2ddb357d71a9045401f8bd0060e3"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jFybY1XBaDtQ+yHmGyFzNJCnLXR9NEv+AReyvosaEXjAl6htb8uHUowxBBpP6syNOjKxVaIBImTFsT/lyfRfDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T01:22:44.277936Z","bundle_sha256":"3dd626193605fb523fbe0f0f799077c6d1138cf3d1c35a5a8401733fe5eb2245"}}