Fractional Calculus & Anomalous Diffusion
D
α
f, MSD ~ t
α
, Lévy walks, waiting-time distributions
Fractional Derivative D
α
of f(x) = x
β
Order α
0.7
Power β
2.0
D
α
x
β
= Γ(β+1)/Γ(β−α+1) · x
β−α
Riemann-Liouville definition via Γ function. α=1 → classical derivative, α=0 → identity.
Mean Squared Displacement: MSD ~ t
α
α (subdiffusion)
0.6
Superdiffusion α
1.6
Normal diffusion:
⟨x²⟩ = 2Dt
(α=1). Subdiffusion α<1: CTRW with power-law waiting times. Superdiffusion α>1: Lévy flights, long jumps.
Lévy Walk Simulation
Lévy index μ
1.8
Steps
200
New Walk
Compare Normal vs Lévy
Lévy walk: step lengths drawn from power-law P(l) ~ l
−μ
. For μ<2, variance diverges → superdiffusion. Waiting times: ψ(t) ~ t
−(1+α)
→ subdiffusion. Anomalous diffusion ubiquitous in biology, finance, physics.
Waiting-Time Distribution
Tail exponent α
0.65
Diffusion Front: Fractional vs Normal
α diffusion
0.6
Time
2.0