# Lévy-Flug & anomale Diffusion

Heavy-Tail-Random-Walks, α-stabile Verteilungen und superdiffusives Skalenverhalten — entdecke nicht-gaußsche Zufallsprozesse, die Tier-Nahrungssuche, Finanzrenditen und turbulenten Transport bestimmen

> Kanonische Seite: https://elysiatools.com/de/visualizations/levy-flight

- **Kategorie:** Math

## Überblick

Interactive visualization of Lévy flights, heavy-tailed random walks, and super-diffusive scaling — exploring the non-Gaussian random processes that govern animal foraging, financial returns, and turbulent transport. Unlike classical Brownian motion (where step lengths follow a Gaussian distribution and the central limit theorem produces √t scaling), Lévy flights draw step lengths from an α-stable distribution with power-law tails P(l) ~ l^(−1−α). When α<2 the variance is infinite: rare but enormous jumps dominate the trajectory, breaking the classical CLT and producing super-diffusive mean squared displacement ⟨r²⟩ ~ t^(2/α). Three simulation modes: Single (one Lévy trajectory animated step by step, with the largest 5% of jumps highlighted in amber to make heavy-tail behavior visually obvious); Compare (Lévy flight versus standard Brownian motion side by side using the same RNG seed, so users can see at any α<2 how Lévy paths exhibit the characteristic 'cluster of small steps + occasional huge jump' pattern absent from Gaussian walks); and Ensemble (200 trajectories evolving in parallel, with median squared displacement plotted on a log-log axis to reveal the t^(2/α) scaling law empirically and a dashed reference line of theoretical slope 2/α). Sampling uses the Chambers-Mallows-Stuck (1976) algorithm: starting from U∈Uniform(−π/2,π/2) and W∈Exp(1), it produces symmetric α-stable samples X = sin(αU)/cos(U)^(1/α) · [cos((1−α)U)/W]^((1−α)/α) (with X = tan(U) at α=1 for the Cauchy case). Each 2D step combines Lévy-distributed length with isotropic uniform direction; for α=2 the construction reduces to a Gaussian random walk that matches Brownian motion in scale. Adjustable parameters: Lévy index α∈[0.5,2.0] in 0.1 steps (0.5=very heavy tail, 1=Cauchy, 1.5=classic Lévy flight, 2=Brownian); drift μ∈[−1,1total trajectory length N∈[200,5000]; animation speed (steps per frame); and ensemble size 50–500. Real-time displays: 2D trajectory canvas with auto-fitting bounding box; position-distribution histogram (x-axis projection, with a Gaussian fit overlay shown only at α≈2 so users can compare empirical fit to theoretical normal); and an MSD-vs-time log-log plot with a least-squares slope estimate and the theoretical 2/α reference line. Live statistics: current step count, current position, largest jump observed, fitted log-log MSD slope (asymptotic regime), expected slope 2/α, and diffusion regime classification (normal / super / strong super-diffusion). Educational sections cover three areas: Theory (α-stable distributions and characteristic function exp(−|σt|^α), heavy tails and power-law step distributions, the Chambers-Mallows-Stuck sampling algorithm); Properties (anomalous MSD scaling t^(2/α) including the linear and quadratic limits at α=2 and α=1, the generalized central limit theorem identifying α-stable laws as the universal attractors for power-law summands, self-similarity with fractal dimension equal to α); and Applications (animal foraging optimality and Viswanathan 1996, fat-tailed financial returns and Mandelbrot 1963 cotton-price work versus Gaussian Black-Scholes underestimation of black swans, turbulent and plasma transport, photon transport in cold-atom systems, and the link to fractional Fokker–Planck equations through continuous-time random walks). Mode-switchable, deterministic with reproducible seed control, and responsive across desktop and mobile viewports. Multi-language support (zh, en, es, fr, de, ru, pt).

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