Relative dispersion of particle pairs in turbulent flow — anomalous superdiffusion
⟨r²(t)⟩ ~ ε·t³ (Richardson 1926) | K(r) ~ ε^(1/3)·r^(4/3) (4/3 law) | ⟨(δr)²⟩ = g·ε·t³
Richardson (1926): Observed that relative diffusivity K(r) ~ r^{4/3} — vastly faster than molecular diffusion (K ~ r⁰). This implies separation ⟨r²⟩ ~ t³, not the t¹ of Brownian motion.
Kolmogorov explanation: In the inertial range (η≪r≪L), velocity differences follow δv(r) ~ (εr)^{1/3} (K41). The relative velocity of a pair separated by r is ~ (εr)^{1/3}, so dr/dt ~ ε^{1/3}r^{1/3} → r(t) ~ (εt³)^{1/2}. The Richardson constant g ≈ 0.5–0.6 measured experimentally.
Simulation: Each particle pair undergoes stochastic evolution with diffusivity K(r) = g·ε^{1/3}·r^{4/3}. The log-log slope of ⟨r²⟩ vs t converges to 3 in the Richardson regime.