{"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_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_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_c_c1d4f07c75cd", "title": "Fokker–Planck & Liouville — probability flow ↔ Fluid probability dynamics — one continuity ", "shelf": "connections", "surface": null, "snippet": "The stochastic instantiation of probability-as-fluid. ... In Hamiltonian phase space, Liouville's theorem gives dρ/dt = 0 — the probability fluid is incompressible.  — a concord the card itself states"}, {"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_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_41cd8b933961", "title": "Fluid probability dynamics — one continuity  ↔ Madelung hydrodynamics — quantum mechanics a", "shelf": "connections", "surface": null, "snippet": "quantum mechanics (Madelung: v = ∇S/m plus a quantum potential — the Schrödinger equation rewritten as fluid dynamics of |ψ|²)  — a concord the card itself states; mined + verified."}, {"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_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_1650c345cc7d", "title": "Pheromones — the molecule is the message ↔ Fluid probability dynamics — one continuity ", "shelf": "connections", "surface": null, "snippet": "Pheromones DIFFUSE down a gradient (the continuity/diffusion equation) and DECAY (the exponential clock) — the fluid and the clock of this whole map, carried in a scent.  — a concord the card itself s"}, {"id": "card_c_d7daafda1482", "title": "Fluid probability dynamics — one continuity  ↔ Diffusion models — the probability-flow ODE ", "shelf": "connections", "surface": null, "snippet": "machine learning (the probability-flow ODE under diffusion models)  — a concord the card itself states; mined + verified."}, {"id": "card_c_37abe9676328", "title": "Fluid probability dynamics — one continuity  ↔ Balancing equations — nothing is created or ", "shelf": "connections", "surface": null, "snippet": "the continuity equation dρ/dt + ∇·J = 0 (current J = ρv) says density is never created or destroyed — it only flows  — a concord the card itself states; mined + verified."}, {"id": "card_src_oeis_a001949", "title": "A001949 — Solutions of a fifth-order probability difference equation.", "shelf": "oeis", "surface": "secular", "snippet": "Solutions of a fifth-order probability difference equation.  First terms: 0, 0, 0, 0, 0, 1, 2, 4, 8, 16, 32, 63, 124, 244, 480, 944, 1856, 3649, 7174, 14104, 27728, 54512, 107168, 210687."}, {"id": "card_src_word_conditional_probability", "title": "conditional probability", "shelf": "dictionary", "surface": "secular", "snippet": "conditional probability: (noun) the probability that an event will occur given that one or more other events have occurred — syn: contingent probability"}, {"id": "card_src_word_contingent_probability", "title": "contingent probability", "shelf": "dictionary", "surface": "secular", "snippet": "contingent probability: (noun) the probability that an event will occur given that one or more other events have occurred — syn: conditional probability"}, {"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_src_word_partial_differential_equation", "title": "partial differential equation", "shelf": "dictionary", "surface": "secular", "snippet": "partial differential equation: (noun) a differential equation involving a functions of more than one variable"}, {"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"}]}