Fractional Diffusion: Subdiffusion & Lévy Subordination

Normal diffusion has ⟨x²⟩ ∝ t. Anomalous diffusion breaks this: subdiffusion (⟨x²⟩ ∝ tᵅ, α<1) arises from power-law waiting times; superdiffusion from Lévy flights with infinite-variance step distributions. The fractional Fokker-Planck equation generalizes the diffusion equation via fractional time/space derivatives, capturing biological crowding, plasma transport, and financial volatility.

Walker trajectories
MSD: ⟨x²⟩ vs time
α (time exponent): 1.00
β (Lévy index): 2.00
N walkers: 20
α=1, β=2: Brownian. α<1: subdiffusion (trapping). β<2: superdiffusion (Lévy flights).