Lévy Flights & Heavy Tails

Lévy flights use power-law step-length distributions P(l) ~ l^(−α−1). Unlike Brownian motion, rare long jumps dominate: the MSD diverges for α≤2, producing superdiffusion.

Lévy Flight (orange)

Brownian Walk (blue)

Lévy index α 1.20

Steps / frame 5

—

MSD Lévy

—

MSD Brownian

—

Max step (Lévy)

—

Steps t

α-Stable Lévy Processes

Step distribution: P(l) ~ l^(−α−1) for large l, α ∈ (0,2]

Characteristic function: φ(k) = exp(−|k|^α)

MSD: ⟨r²⟩ ~ t^(2/α) (diverges for α < 2)

α=2: Gaussian (Brownian) | α=1: Cauchy

Applications: foraging, finance, turbulent diffusion