Random walk aggregation · fractal dimension ~1.71 · Witten-Sander model
Particles: 1
Radius: 0
Est. dim D_f: —
Density: —
Diffusion-Limited Aggregation (DLA) — Witten & Sander (1981): particles
diffuse randomly until they contact the growing cluster and stick permanently.
This minimal model generates fractal structures with remarkable universality.
The fractal dimension D_f ≈ 1.71 in 2D (exact value unknown analytically).
Mass within radius r scales as M(r) ~ r^{D_f}. The branched, tip-dominated
morphology arises because tips capture more diffusing particles (Laplacian growth).
The cluster solves ∇²u=0 outside with u=0 on the surface — the growth probability
equals the harmonic measure.
Reducing sticking probability p causes particle averaging and smoother, denser clusters
with higher D_f → approaching D_f=2 (compact) for small p. DLA describes:
electrodeposition, mineral dendrites, dielectric breakdown, viscous fingering.