Heavy-tailed steps, anomalous diffusion, and the statistics of foraging animals
Parameters
Steps:0
MSD α slope:—
Expected: α=—
A Lévy flight uses step lengths drawn from a power-law distribution P(l) ~ l−(1+α). For α < 2, the variance is infinite — the walker can make arbitrarily long jumps. MSD grows as ⟨r²⟩ ~ tγ with γ > 1 (superdiffusion). When α → 2, standard Brownian motion is recovered.
Real-world Lévy foraging: albatrosses, shark hunting paths, human mobility (mobile phone data), earthquake aftershock patterns. The Lévy strategy is theoretically optimal when prey is sparse and randomly distributed (Viswanathan et al 1999, Nature).