{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:XFGJNFPPN2RWJUOF3AG2N64FEO","short_pith_number":"pith:XFGJNFPP","canonical_record":{"source":{"id":"1802.05926","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2018-02-16T13:36:58Z","cross_cats_sorted":["math.AP","math.DS","math.MP"],"title_canon_sha256":"21d54ee4b51839b4ba4563200aff738a31f46ed573af19404c48480efb63a760","abstract_canon_sha256":"0a9c18410e5e43bcc010992ba24d9905ed47682e7b69d115fc262df60c17f2b2"},"schema_version":"1.0"},"canonical_sha256":"b94c9695ef6ea364d1c5d80da6fb8523bd0332a1a4b007583aa355d8d0d7b756","source":{"kind":"arxiv","id":"1802.05926","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1802.05926","created_at":"2026-05-18T00:23:10Z"},{"alias_kind":"arxiv_version","alias_value":"1802.05926v1","created_at":"2026-05-18T00:23:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1802.05926","created_at":"2026-05-18T00:23:10Z"},{"alias_kind":"pith_short_12","alias_value":"XFGJNFPPN2RW","created_at":"2026-05-18T12:33:01Z"},{"alias_kind":"pith_short_16","alias_value":"XFGJNFPPN2RWJUOF","created_at":"2026-05-18T12:33:01Z"},{"alias_kind":"pith_short_8","alias_value":"XFGJNFPP","created_at":"2026-05-18T12:33:01Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:XFGJNFPPN2RWJUOF3AG2N64FEO","target":"record","payload":{"canonical_record":{"source":{"id":"1802.05926","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2018-02-16T13:36:58Z","cross_cats_sorted":["math.AP","math.DS","math.MP"],"title_canon_sha256":"21d54ee4b51839b4ba4563200aff738a31f46ed573af19404c48480efb63a760","abstract_canon_sha256":"0a9c18410e5e43bcc010992ba24d9905ed47682e7b69d115fc262df60c17f2b2"},"schema_version":"1.0"},"canonical_sha256":"b94c9695ef6ea364d1c5d80da6fb8523bd0332a1a4b007583aa355d8d0d7b756","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:23:10.488536Z","signature_b64":"x2Uopq4LjQBit3E9JOC0zx0LWHcQaj+uSI/4P4JyLQEf3QdySr5IkM61rXCJnVm4t5UJf6qvPbdNI5B1/Y5+BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b94c9695ef6ea364d1c5d80da6fb8523bd0332a1a4b007583aa355d8d0d7b756","last_reissued_at":"2026-05-18T00:23:10.487690Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:23:10.487690Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1802.05926","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":"2026-05-18T00:23:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1iQOaL42UJqjoaO0uUiKhdvFTITTNZKaCSnfWTyGl5pleug/fxunhDm/XJZbLr+XkJ/akOqkCTYaA2UVPri8BQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T19:35:00.171726Z"},"content_sha256":"0a798bd9447ffa7cde2e7fd4ebbab23101099c601f02a3b771bda4690d441710","schema_version":"1.0","event_id":"sha256:0a798bd9447ffa7cde2e7fd4ebbab23101099c601f02a3b771bda4690d441710"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:XFGJNFPPN2RWJUOF3AG2N64FEO","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Microscopic solutions of the Boltzmann-Enskog equation in the series representation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.DS","math.MP"],"primary_cat":"math-ph","authors_text":"Anton Trushechkin, Mario Pulvirenti, Sergio Simonella","submitted_at":"2018-02-16T13:36:58Z","abstract_excerpt":"The Boltzmann-Enskog equation for a hard sphere gas is known to have so called microscopic solutions, i.e., solutions of the form of time-evolving empirical measures of a finite number of hard spheres. However, the precise mathematical meaning of these solutions should be discussed, since the formal substitution of empirical measures into the equation is not well-defined. Here we give a rigorous mathematical meaning to the microscopic solutions to the Boltzmann-Enskog equation by means of a suitable series representation."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.05926","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":"2026-05-18T00:23:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vHIGO2i3d5N3nrUn8MK/jMqnIFBkVBsr0O/TpnZF/8WthfUgvYhb6h3wiO2426qi71wzdBdqwZh74moAgJF4Cg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T19:35:00.172075Z"},"content_sha256":"d4c2372cb738c43f20a4b407a8345239345104633975b559264add339d905c51","schema_version":"1.0","event_id":"sha256:d4c2372cb738c43f20a4b407a8345239345104633975b559264add339d905c51"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XFGJNFPPN2RWJUOF3AG2N64FEO/bundle.json","state_url":"https://pith.science/pith/XFGJNFPPN2RWJUOF3AG2N64FEO/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XFGJNFPPN2RWJUOF3AG2N64FEO/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":"2026-06-04T19:35:00Z","links":{"resolver":"https://pith.science/pith/XFGJNFPPN2RWJUOF3AG2N64FEO","bundle":"https://pith.science/pith/XFGJNFPPN2RWJUOF3AG2N64FEO/bundle.json","state":"https://pith.science/pith/XFGJNFPPN2RWJUOF3AG2N64FEO/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XFGJNFPPN2RWJUOF3AG2N64FEO/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:XFGJNFPPN2RWJUOF3AG2N64FEO","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":"0a9c18410e5e43bcc010992ba24d9905ed47682e7b69d115fc262df60c17f2b2","cross_cats_sorted":["math.AP","math.DS","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2018-02-16T13:36:58Z","title_canon_sha256":"21d54ee4b51839b4ba4563200aff738a31f46ed573af19404c48480efb63a760"},"schema_version":"1.0","source":{"id":"1802.05926","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1802.05926","created_at":"2026-05-18T00:23:10Z"},{"alias_kind":"arxiv_version","alias_value":"1802.05926v1","created_at":"2026-05-18T00:23:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1802.05926","created_at":"2026-05-18T00:23:10Z"},{"alias_kind":"pith_short_12","alias_value":"XFGJNFPPN2RW","created_at":"2026-05-18T12:33:01Z"},{"alias_kind":"pith_short_16","alias_value":"XFGJNFPPN2RWJUOF","created_at":"2026-05-18T12:33:01Z"},{"alias_kind":"pith_short_8","alias_value":"XFGJNFPP","created_at":"2026-05-18T12:33:01Z"}],"graph_snapshots":[{"event_id":"sha256:d4c2372cb738c43f20a4b407a8345239345104633975b559264add339d905c51","target":"graph","created_at":"2026-05-18T00:23:10Z","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 Boltzmann-Enskog equation for a hard sphere gas is known to have so called microscopic solutions, i.e., solutions of the form of time-evolving empirical measures of a finite number of hard spheres. However, the precise mathematical meaning of these solutions should be discussed, since the formal substitution of empirical measures into the equation is not well-defined. Here we give a rigorous mathematical meaning to the microscopic solutions to the Boltzmann-Enskog equation by means of a suitable series representation.","authors_text":"Anton Trushechkin, Mario Pulvirenti, Sergio Simonella","cross_cats":["math.AP","math.DS","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2018-02-16T13:36:58Z","title":"Microscopic solutions of the Boltzmann-Enskog equation in the series representation"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.05926","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":"sha256:0a798bd9447ffa7cde2e7fd4ebbab23101099c601f02a3b771bda4690d441710","target":"record","created_at":"2026-05-18T00:23:10Z","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":"0a9c18410e5e43bcc010992ba24d9905ed47682e7b69d115fc262df60c17f2b2","cross_cats_sorted":["math.AP","math.DS","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2018-02-16T13:36:58Z","title_canon_sha256":"21d54ee4b51839b4ba4563200aff738a31f46ed573af19404c48480efb63a760"},"schema_version":"1.0","source":{"id":"1802.05926","kind":"arxiv","version":1}},"canonical_sha256":"b94c9695ef6ea364d1c5d80da6fb8523bd0332a1a4b007583aa355d8d0d7b756","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b94c9695ef6ea364d1c5d80da6fb8523bd0332a1a4b007583aa355d8d0d7b756","first_computed_at":"2026-05-18T00:23:10.487690Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:23:10.487690Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"x2Uopq4LjQBit3E9JOC0zx0LWHcQaj+uSI/4P4JyLQEf3QdySr5IkM61rXCJnVm4t5UJf6qvPbdNI5B1/Y5+BQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:23:10.488536Z","signed_message":"canonical_sha256_bytes"},"source_id":"1802.05926","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0a798bd9447ffa7cde2e7fd4ebbab23101099c601f02a3b771bda4690d441710","sha256:d4c2372cb738c43f20a4b407a8345239345104633975b559264add339d905c51"],"state_sha256":"ee4a5402dd31a618298adc7dd7b2876e0a240940d14bf321cb56834a2dc1befa"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HF0cWfHTBPQe7eYjH/wYVN5ytvTCQE/8zrb6DKO1sv/HrLx0ezBRMcMk66cFOY0A4x4Bpn6b5aeQoPTeh/nUDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-04T19:35:00.174089Z","bundle_sha256":"c7eb718890ba215e700fd8ad05b7efd16e102f57bf42dec16d6e6aa54a98c6fe"}}