Correlated Random Walk — Persistent Motion

Ballistic to diffusive crossover: how persistence time controls transport
⟨r²(t)⟩ = 2v²τ²[t/τ − (1 − e^{−t/τ})] → v²t² (t≪τ) → 2Dt (t≫τ)

Walk Parameters

ρ = 0.85
v = 2.0
6 walkers
3

Statistics

Steps taken0
Persistence time τ—
Diffusion coeff D = v²τ—
⟨r²⟩ measured—
⟨r²⟩ predicted—
Crossover time τ—
Correlated walk
Simple random walk
t² (ballistic)
t (diffusive)

In a persistent random walk, each step is correlated with the previous direction. The correlation ρ gives persistence time τ = −1/ln(ρ). At short times t ≪ τ, motion is ballistic (⟨r²⟩ ~ t²) because direction hasn't randomized yet. At long times t ≫ τ, it's diffusive (⟨r²⟩ ~ t) with D = v²τ. This models bacterial swimming (run-and-tumble), active matter, and anomalous diffusion in complex systems.