{"query": "Madelung hydrodynamics — quantum mechanics as a probability", "count": 20, "results": [{"id": "card_n_810d857d886e", "title": "Madelung hydrodynamics — quantum mechanics as a probability fluid", "shelf": "science", "surface": "secular", "snippet": "Write ψ = √ρ·e^(iS/ħ) and the Schrödinger equation becomes fluid dynamics of the probability\ndensity ρ = |ψ|²: a continuity equation with velocity v = ∇S/m, plus an Euler equation carrying a\n'quantum "}, {"id": "card_c_576d28632055", "title": "Madelung hydrodynamics instantiates Fluid probability dynamics", "shelf": "connections", "surface": null, "snippet": "Quantum mechanics rewritten as the fluid dynamics of |ψ|²."}, {"id": "card_n_8530c72a2201", "title": "Fluid probability dynamics — one continuity equation across physics, geometry, and ML", "shelf": "science", "surface": "secular", "snippet": "One equation wears many clothes. Probability is a conserved fluid: the continuity equation\ndρ/dt + ∇·J = 0 (current J = ρv) says density is never created or destroyed — it only flows. The\nsame skeleto"}, {"id": "card_n_041763374eca", "title": "Fokker–Planck & Liouville — probability flow in statistical mechanics", "shelf": "science", "surface": "secular", "snippet": "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 cur"}, {"id": "card_c_3cc9f3e8e74f", "title": "Fokker–Planck & Liouville instantiates Fluid probability dynamics", "shelf": "connections", "surface": null, "snippet": "Statistical mechanics is the stochastic instance of the probability-continuity equation."}, {"id": "card_c_e655b3fcef54", "title": "Optimal transport instantiates Fluid probability dynamics", "shelf": "connections", "surface": null, "snippet": "The geometry of moving probability — Fokker–Planck as a Wasserstein gradient flow."}, {"id": "card_c_e510a4e8b7f4", "title": "Diffusion models instantiate Fluid probability dynamics", "shelf": "connections", "surface": null, "snippet": "The probability-flow ODE is the same continuity equation with a learned velocity."}, {"id": "card_n_2013d540edf1", "title": "The Pressure Architecture — Leverage and Fulcrums", "shelf": "domains", "surface": "secular", "snippet": "Every offensive has a paired defensive.\nIf they have a lever, shorten it.\nControl the fulcrum to define the lever.\nMechanics emerge from positioning.\n\nLeverage is not separate from position — it IS po"}, {"id": "card_n_2be7f1c3f6e9", "title": "Diffusion models — the probability-flow ODE behind generative AI", "shelf": "science", "surface": "secular", "snippet": "Score-based generative models noise data toward a Gaussian and learn to reverse it. The\n'probability-flow ODE' is the deterministic fluid whose time-marginals match the noising SDE — the\nsame continui"}, {"id": "card_n_0a09f138d552", "title": "Replace a Faucet Washer", "shelf": "maker", "surface": "secular", "snippet": "A leaky faucet costs gallons a day; the fix is a 10-cent washer. Teach a teenager this once and they save themselves money for life."}, {"id": "card_n_ab76428e9fa5", "title": "Make a Newspaper Kite", "shelf": "maker", "surface": "secular", "snippet": "A diamond kite from newsprint and thin sticks. Lifts on the lightest breeze. Free."}, {"id": "card_n_1e2210fa1f19", "title": "Build a Bird Feeder from Scrap Wood", "shelf": "maker", "surface": "secular", "snippet": "A simple platform feeder with a roof. Birds find it within a day. Watch them through the kitchen window all winter."}, {"id": "card_n_f1e483b64f08", "title": "Build a Soap-Box Car", "shelf": "maker", "surface": "secular", "snippet": "Four wheels, a plank, a steering rope, two parents — the classic American gravity racer. Heir to the Soap Box Derby."}, {"id": "card_c_3437884ad1bb", "title": "Fluid probability dynamics related to the spectral (Fourier) lens", "shelf": "connections", "surface": null, "snippet": "Fourier eigenmodes of the Laplacian are exactly how the diffusion/continuity operator is solved."}, {"id": "card_n_ea9e3f380756", "title": "The primes and the nucleus — one statistical fingerprint", "shelf": "science", "surface": "secular", "snippet": "The night's deepest rhyme, and it is real, published, and unexplained. Montgomery (1973)\nand Dyson noticed that the SPACINGS between the Riemann zeta zeros follow the same distribution\nas the eigenval"}, {"id": "card_c_108666f40f9f", "title": "Fluid probability dynamics resonates with Fluid dynamics of the axes", "shelf": "connections", "surface": null, "snippet": "Same conserved-flow skeleton, read on the map's own axes — RESONANCE, a map-aid not a proof."}, {"id": "card_c_aba992724479", "title": "Pheromones diffuse — the continuity equation, in scent", "shelf": "connections", "surface": null, "snippet": "A scent gradient is a probability fluid flowing; the ant reads the density."}, {"id": "card_n_cda0b1e1a364", "title": "Fusion — why the stars shine", "shelf": "science", "surface": "secular", "snippet": "Four hydrogens weigh more than one helium; the difference is sunlight. Sealed from\nthe measured masses: 4·m(H) − m(He-4) = 0.0287 u → 26.73 MeV per helium forged: https://narrowhighway.com/s/8a5e89ab1"}, {"id": "card_c_663f60b2cc20", "title": "Schumann resonance resonates with Fluid probability dynamics", "shelf": "connections", "surface": null, "snippet": "Both are eigenmode stories — standing waves of a bounded medium; the recurrence is RESONANCE, a map-aid, never a proof of unity."}, {"id": "card_n_6e7a5b14e556", "title": "The periodic table — the map of the elements", "shelf": "science", "surface": "secular", "snippet": "118 elements, ordered by proton count, and their properties RECUR in periods — the\nsingle most predictive table in science. Sealed in-domain: carbon is element 6, symbol C\n(https://narrowhighway.com/s"}]}