Lévy Walk · Anomalous Diffusion

Heavy-tail step lengths · MSD ~ tα with α>1

Lévy Exponent

α (MSD exponent): —
Mean |r|²: —
Step: 0
log MSD vs log t
A Lévy walk draws step lengths from a power-law distribution P(ℓ) ~ ℓ-μ. For 1 < μ < 2 (heavy tail, infinite variance), the mean squared displacement grows superdiffusively: MSD ~ tα with α = (3-μ) > 1. For μ > 3, ordinary diffusion (α=1) recovers. Lévy walks model animal foraging, light in random media, and human mobility patterns.