Superdiffusive

Superdiffusive describes diffusion processes in which the mean squared displacement grows faster than linearly with time. Formally, ⟨x^2(t)⟩ ∼ C t^α for long times with α>1. Normal diffusion corresponds to α=1, while α=2 represents ballistic motion. Subdiffusion, by contrast, has α<1.

Mechanisms leading to superdiffusion include heavy-tailed jump distributions such as Lévy flights, persistent or correlated motion as in fractional Brownian motion with H>1/2, and certain active-transport or persistent-flow scenarios. Lévy flights yield occasional very long jumps that drive rapid dispersion; Lévy walks enforce finite velocity and can modify the scaling in practical time windows.

Superdiffusion appears in physical, biological, and ecological contexts, including turbulent diffusion, plasma transport, intracellular motility driven by motor proteins, and foraging patterns in animals. It also arises in models of active matter, where interactions generate persistent motion and long-range correlations.

Measuring superdiffusion typically involves estimating the ensemble-averaged MSD from many trajectories or from long single trajectories. Some models exhibit non-ergodic behavior, in which time and ensemble averages do not coincide, complicating interpretation.

In fractional Brownian motion, the MSD scales as ⟨x^2(t)⟩ ∝ t^{2H}, so α=2H with H>0.5 giving superdiffusion. In contrast, processes with infinite-variance jumps produce ill-defined MSDs. Distinguishing mechanisms requires careful analysis of trajectories and scaling behavior.