{"paper":{"title":"Linearization Principle: The Geometric Origin of Nonlinear Fokker-Planck Equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"The Linearization Principle derives nonlinear Fokker-Planck equations geometrically by keeping the drift term linear in probability density.","cross_cats":["math-ph","math.MP"],"primary_cat":"cond-mat.stat-mech","authors_text":"Hiroki Suyari","submitted_at":"2026-03-01T21:31:32Z","abstract_excerpt":"Anomalous diffusion and power-law distributions are observed in various complex systems. To provide a consistent dynamical foundation for these phenomena, we present a geometric derivation of the nonlinear Fokker-Planck equation by introducing the Linearization Principle directly at the dynamical stage. By identifying the generalized chemical potential as the natural dynamical ansatz, we construct a general thermodynamic framework where the drift term remains linear in the probability density, preserving the standard form of the Einstein relation. Within this framework, we show that the $q$-de"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"By identifying the generalized chemical potential as the natural dynamical ansatz, we construct a general thermodynamic framework where the drift term remains linear in the probability density, preserving the standard form of the Einstein relation. Within this framework, we show that the q-deformed geometry... exhibits a fundamental duality between the dynamic index q and the thermodynamic index 2-q.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The Linearization Principle itself, introduced directly at the dynamical stage as the central geometric rule that forces the drift to stay linear while allowing nonlinear diffusion.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A geometric Linearization Principle derives nonlinear Fokker-Planck equations that preserve linear drift and Einstein relation while producing q-Gaussian equilibria minimizing a 2-q entropy functional.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The Linearization Principle derives nonlinear Fokker-Planck equations geometrically by keeping the drift term linear in probability density.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"52556522ddc80f1154e2735a84f7c1743e5cd49bb18ec26d9bc438c12a975859"},"source":{"id":"2603.01278","kind":"arxiv","version":3},"verdict":{"id":"8bdd8f03-b74e-4a57-ab02-03a89099cdd6","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T17:42:35.692756Z","strongest_claim":"By identifying the generalized chemical potential as the natural dynamical ansatz, we construct a general thermodynamic framework where the drift term remains linear in the probability density, preserving the standard form of the Einstein relation. Within this framework, we show that the q-deformed geometry... exhibits a fundamental duality between the dynamic index q and the thermodynamic index 2-q.","one_line_summary":"A geometric Linearization Principle derives nonlinear Fokker-Planck equations that preserve linear drift and Einstein relation while producing q-Gaussian equilibria minimizing a 2-q entropy functional.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The Linearization Principle itself, introduced directly at the dynamical stage as the central geometric rule that forces the drift to stay linear while allowing nonlinear diffusion.","pith_extraction_headline":"The Linearization Principle derives nonlinear Fokker-Planck equations geometrically by keeping the drift term linear in probability density."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2603.01278/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":3,"snapshot_sha256":"f683d09d715f4652a096f2eef3f7319d174752d7d644fade326100f4406dc84d"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}