Lévy Walk: Anomalous Superdiffusion

⟨r²(t)⟩ ~ t^α with α > 1 from power-law step lengths

Lévy index μ: 1.5 Walkers: 5

Steps

⟨r²⟩ (log)

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Diffusion exp α

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Max step

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A Lévy walk combines power-law step lengths P(l) ~ l^-μ with finite velocity, so time scales with distance. For 1<μ<2 the mean-square displacement grows as ⟨r²⟩ ~ t^α with α = 4-μ > 2 — superdiffusion. Lévy walks model animal foraging (Brownian when food is dense, Lévy when sparse), human mobility, light scattering in Lévy glasses, and blinking quantum dots.