{"query": "Fluid probability dynamics — one continuity equation across", "count": 20, "results": [{"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_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_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_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_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_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_n_ff27b07c4577", "title": "Fluid dynamics as the behavior of the axes", "shelf": "concepts", "surface": "secular", "snippet": "Matt: 'Fluid dynamics as behavior of axes? something along that line.' The axes are not static — they FLOW and redistribute as the map grows. Fluid dynamics may describe their behavior: a turbulent fl"}, {"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_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_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_d59ca677ea24", "title": "Optimal transport — the geometry of moving probability", "shelf": "science", "surface": "secular", "snippet": "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 F"}, {"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_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_1b6fc0a79c10", "title": "Ignatius to the Magnesians VII", "shelf": "patristics", "surface": "witness", "snippet": "As therefore the Lord did nothing without the Father, being united to Him, neither by Himself nor by the apostles, so neither do ye anything without the bishop and presbyters. Neither endeavour that a"}, {"id": "card_n_88ac7219a4a9", "title": "Laplace — the map's dynamics (decay/growth), beyond the steady spectrum", "shelf": "concepts", "surface": "secular", "snippet": "Matt: 'laplace transform ... missing.' Fourier gives the STEADY spectrum (the imaginary axis); Laplace adds the REAL axis s=sigma+i*omega — decay/growth RATES. The map's eigenvalue spectrum decays EXP"}, {"id": "card_n_137e090d24e8", "title": "Aurelius, Meditations §aur_04_xxxiii: Ever consider and think upon the world as being but one living substance, and...", "shelf": "classics", "surface": "secular", "snippet": "Ever consider and think upon the world as being but one living substance, and having but one soul, and how all things in the world, are terminated into one sensitive power; and are done by one general"}, {"id": "card_n_53c81d366707", "title": "The Chalcedonian Definition", "shelf": "codex", "surface": "witness", "snippet": "Following the holy fathers, we all with one accord teach men to acknowledge one and the same Son, our Lord Jesus Christ, at once complete in Godhead and complete in manhood, truly God and truly man, c"}, {"id": "card_n_edf79a6e0295", "title": "Easton: Zebah", "shelf": "dictionary", "surface": "secular", "snippet": "Man-killer, or sacrifice, one of the two kings who led the vast host of the Midianites who invaded the land of Israel, and over whom Gideon gained a great and decisive victory (Judg. 8). Zebah and Zal"}, {"id": "card_n_16e019f825c8", "title": "Easton: Adullam", "shelf": "dictionary", "surface": "secular", "snippet": "One of the royal cities of the Canaanites, now ‘Aid-el-ma (Josh. 12:15; 15:35). It stood on the old Roman road in the valley of Elah (q.v.), which was the scene of David’s memorable victory over Golia"}, {"id": "card_n_51ecbfa0bb5a", "title": "The radio spectrum — one law from the cavity to the eye", "shelf": "science", "surface": "secular", "snippet": "Walk the dial upward and nothing changes but the number: λ·f = c the whole\nway. Twelve rungs sealed in ONE receipt: https://narrowhighway.com/s/d6211f4efb2518c6de145decf57f27b131a01c24ce7841853f395b7b"}]}