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Anomalous Diffusion — Sub and Superdiffusion
Normal diffusion: ⟨x²⟩ ∝ t. Anomalous: ⟨x²⟩ ∝ t^α — from trapped particles to Lévy flights
Particle Trajectories
MSD = ⟨x²(t)⟩ (log-log)
Mode
Normal (α=1)
Subdiffusion (α<1)
Superdiffusion / Lévy (α>1)
All Three
α exponent:
1.00
N walkers:
20
Run
Reset
Slope α = —
Normal (Brownian): steps ~ Gaussian, ⟨x²⟩ = 2Dt (α=1). Subdiffusion: particle gets trapped (power-law waiting times), ⟨x²⟩ ~ t^α, α<1. Protein diffusion in cytoplasm. Superdiffusion (Lévy): occasional huge jumps (power-law step sizes), ⟨x²⟩ ~ t^α, α>1. Albatross foraging, human travel. Log-log slope of MSD gives α — compare to reference lines.