Superdiffusiv

Superdiffusiv, or superdiffusion, refers to diffusion processes in which the mean squared displacement ⟨x^2(t)⟩ grows faster than linearly with time, typically as ⟨x^2(t)⟩ ∝ t^α with α > 1. This contrasts with normal diffusion (α = 1) and subdiffusion (α < 1). Superdiffusive behavior often arises from long-range correlations, persistent motion, or occasional very long jumps.

Common models include Lévy flights, which have heavy-tailed jump length distributions, and Lévy walks, which constrain velocity but still yield superdiffusion. Fractional Brownian motion with Hurst exponent H > 1/2 also produces superdiffusive scaling, since ⟨x^2(t)⟩ ∝ t^{2H}.

In physical and biological systems, superdiffusion describes transport and exploration patterns that spread more rapidly than ordinary diffusion. It appears in turbulent fluids and plasmas, atmospheric pollutant transport, intracellular movement of organelles, and foraging patterns in some animals. It also helps explain non-Gaussian displacement distributions with fat tails observed in experiments.

Observables include the scaling of the mean squared displacement with time, ⟨x^2(t)⟩ ∝ t^α (α>1), and the presence of heavy tails in the displacement distribution. Time-averaged and ensemble-averaged measures can diverge in certain models, reflecting weak ergodicity breaking.

See also: anomalous diffusion; Lévy flight; fractional Brownian motion.