experiment 024 · fractal growth · stochastic physics

Diffusion-Limited Aggregation

Release a particle far outside a seed. Let it wander at random — equal probability in each of four directions, no memory, no destination. The moment it brushes the cluster it freezes permanently. Repeat. The result is not chaos but a fractal crystal: a geometry that emerges wherever slow diffusion meets sharp adhesion — in lightning, coral, frost, and the branching of your lungs.

Dynamics

sticking probability 1.00

active walkers 80

Appearance

colour palette

Seed

seed geometry

Click the canvas to drop an extra seed and spawn local walkers around it.

Metrics

particles stuck 1

cluster radius 1.0

fractal dim D_f —

active walkers 80

Two rules, infinite complexity

The algorithm is disarmingly simple. A particle is released at random on a circle whose radius exceeds the cluster's current extent. It performs a lattice random walk — equivalent to Brownian motion in the continuum limit — until it finds a neighbour that already belongs to the cluster. At that moment it adheres and becomes part of the structure forever. A replacement particle is immediately released and the process repeats.

What emerges is not a compact disk. The cluster develops arms and branches because protruding tips intercept diffusing particles more readily than concave fjords — a geometric instability that amplifies small fluctuations into the characteristic dendritic silhouette.

Laplacian growth

DLA is the discrete realisation of Laplacian growth: an interface whose local velocity is proportional to the gradient of a harmonic field (here, particle concentration). The field obeys ∇²φ = 0 everywhere outside the cluster with φ = 0 on the cluster surface and φ → 1 at infinity. Tips of the cluster live in regions of high ∂φ/∂n and therefore grow fastest, sealing the fjords behind them. The same equation governs viscous fingering in Hele-Shaw cells, electrostatic discharge, and solidification fronts.

Fractal dimension

A compact disk of radius R contains πR² particles. A DLA cluster with the same radius holds only R^D_f particles, where D_f ≈ 1.71 in two dimensions — comfortably between a line (D=1) and a filled plane (D=2). The readout above estimates it live from D_f = log(N) / log(R). You will see it settle near 1.7 as the cluster matures, regardless of the random seed or initial geometry.

Sticking probability

With p < 1, a particle touching the cluster rebounds with probability 1 − p. Lower p smooths the branches: a particle can explore the concave recesses it would otherwise miss, filling in some gaps and increasing the effective dimension toward 2. At p → 0 you approach Eden growth, which produces a compact, roughly circular cluster. Tuning p is therefore a dial between fractal and Euclidean geometry.

In the wild

LightningDielectric breakdown traces the gradient of the electrostatic field through air, branching wherever local field strength peaks.

ElectrodepositionCopper ions drift and diffuse in solution, adhering to a cathode and building metallic dendrites that can short-circuit a battery.

Coral & lichenColonial organisms grow toward nutrient gradients, their tips intercepting supply while sheltered valleys starve — Laplacian selection without the equations.

Snowflake armsWater vapour deposits preferentially on protruding ice crystal facets; each arm independently traces the same ambient humidity field.

River deltasSediment carried by a decelerating river bifurcates along diffusion-dominated pathways, building fractal fans at the sea.

Lung branchingBronchial trees and vascular networks optimise surface area under diffusion-limited transport — the same trade-off, solved by evolution.

Numerical notes

The walker count controls throughput: more walkers grow the cluster faster but each covers fewer steps per frame. A jump optimisation collapses the many wasted steps a particle would otherwise spend diffusing far from the cluster — when a walker's distance from the origin exceeds the cluster radius by more than a threshold, it leaps to a position just outside the cluster boundary rather than walking step by step from afar. This accelerates growth without biasing where particles arrive on the cluster surface. The grid is 401 × 401 cells; particle colours are indexed to their arrival order so inner rings always reflect earlier deposition.

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