Anomalous Diffusion

Fractional Brownian motion (Mandelbrot & Van Ness, 1968) generalizes Brownian motion using the Hurst exponent H. Normal diffusion: H=0.5, MSD ~ t. Superdiffusion (H>0.5): persistent motion, MSD grows faster — seen in active matter. Subdiffusion (H<0.5): anti-persistent, MSD grows slower — seen in crowded cells, gels. The increments have long-range correlations: C(k) ~ |k|^2H−2.