{"id": "card_n_d59ca677ea24", "kind": "note", "title": "Optimal transport — the geometry of moving probability", "body": "Benamou–Brenier: the distance between two distributions is the least kinetic energy of a\nfluid that carries one to the other, subject to the continuity equation. Jordan–Kinderlehrer–Otto\n(1998): the Fokker–Planck equation is the gradient flow of free energy F = ∫ρ log ρ + ∫Uρ in this\nWasserstein geometry — diffusion is steepest descent of entropy.", "source": {"label": "Concordance assay — 2026-07-08", "url": "https://narrowhighway.com/s/8a3d0e4b4c8af33851fc97658b7af1f9631e8a770708a245234f2226f076d694", "ref": "optimal_transport", "authority_tier": "engine_derived"}, "shelf": "science", "box": "fluid_probability_dynamics", "bands": ["optimal transport", "wasserstein", "benamou-brenier", "jko", "gradient flow"], "connections": [{"to_card_id": "card_n_8530c72a2201", "relationship": "instantiates", "origin": "assay_2026_07_08"}, {"to_card_id": "card_k_spine_created_order", "relationship": "member_of", "evidence": "a member of the science shelf in the keeping"}], "author": "engine", "created_at": "2026-07-09T21:52:14.887408+00:00", "updated_at": "2026-07-09T21:52:14.887408+00:00", "visibility": "public", "lifecycle_stage": "public", "volatility": "permanent", "surface": "secular", "metrics": {"paperclips_count": 0, "helpful_count": 0, "not_helpful_count": 0, "cite_count": 0, "walks_through_count": 0, "flagged_count": 0}, "source_hash": "70aa1be583850d72dcb5c683f4fd3b239ef301a182df789c93b54f70c6f51351", "presentation": {"glyph": "•", "kind_label": "fluid probability dynamics", "by": "Concordance assay — 2026-07-08", "authority": "engine_derived", "posted": "", "standing": "", "link": {"url": "https://narrowhighway.com/s/8a3d0e4b4c8af33851fc97658b7af1f9631e8a770708a245234f2226f076d694", "host": "narrowhighway.com", "provider": "", "titled": "Optimal transport — the geometry of moving probability", "waybill_line": "", "reach": "", "embed": ""}}, "neighbors": [{"id": "card_n_8530c72a2201", "title": "Fluid probability dynamics — one continuity equation across physics, geometry, and ML", "relationship": "instantiates", "why": "", "href": "/card/card_n_8530c72a2201", "resolved": true}, {"id": "card_k_spine_created_order", "title": "The created order — the science of the floor", "relationship": "on the shelf of", "why": "a member of the science shelf in the keeping", "href": "/card/card_k_spine_created_order", "resolved": true}]}