{"id": "card_n_041763374eca", "kind": "note", "title": "Fokker–Planck & Liouville — probability flow in statistical mechanics", "body": "The stochastic instantiation of probability-as-fluid. Fokker–Planck sets the current\nJ = μρ − D∇ρ (drift + diffusion); its steady state is the Boltzmann distribution ρ ∝ e^(−U/kT),\nwhich makes the current vanish and whose variance equals the temperature (sealed: https://narrowhighway.com/s/8a3d0e4b4c8af33851fc97658b7af1f9631e8a770708a245234f2226f076d694). In\nHamiltonian phase space, Liouville's theorem gives dρ/dt = 0 — the probability fluid is\nincompressible.", "source": {"label": "Concordance assay — 2026-07-08", "url": "https://narrowhighway.com/s/8a3d0e4b4c8af33851fc97658b7af1f9631e8a770708a245234f2226f076d694", "ref": "fokker_planck", "authority_tier": "engine_derived"}, "shelf": "science", "box": "fluid_probability_dynamics", "bands": ["fokker-planck", "liouville", "boltzmann", "statistical mechanics", "diffusion", "entropy"], "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": "b6a01361fe491f63f5aceb8deb6e855f4640f9d5e465be56a6ae824cca7321c8", "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": "Fokker–Planck & Liouville — probability flow in statistical mechanics", "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}]}