Fractional Calculus — Anomalous Diffusion & Subdiffusion

Mean squared displacement ⟨x²⟩ ~ t^α with α ≠ 1

Diffusion Type

Statistics

⟨x²⟩ (measured) = 0.00
Predicted t^α = 0.00
Time elapsed t = 0.00
Regime: Subdiffusion
α<1: subdiff (traps) | α=1: normal | α>1: superdiff (Lévy)
Physics: Normal Brownian diffusion yields ⟨x²⟩ = 2Dt (α=1). Anomalous diffusion occurs in disordered media, biological cells, and crowded environments: ⟨x²⟩ ~ 2D_α t^α. Subdiffusion (α<1) arises from long waiting times in traps (CTRW model, Montroll-Weiss) — modeled by a fractional derivative ∂^α u/∂t^α = D_α ∂²u/∂x². Superdiffusion (α>1) comes from long jumps (Lévy flights). The fractional Fokker-Planck equation generalizes classical diffusion. MSD log-log slope = α is the key diagnostic.