Lévy Flight: Truncated Stable Distribution

Lévy flights with stability index α<2 make superdiffusive jumps. Truncating the tail at scale λ yields tempered stable motion with finite moments — a crossover from anomalous to Gaussian diffusion at large scales.

Parameters

α = 2: Brownian motion (Gaussian)
α < 2: Heavy-tailed jumps
α = 1: Cauchy process

Truncation cuts jumps larger than λ,
creating finite variance.

MSD ~ t2/α (untruncated)
MSD ~ t (truncated, large t)

Jump size distribution shown below
on log-log scale.