What is diffusion-limited aggregation (DLA)?

DLA is a fractal growth model where particles perform Brownian random walks until they contact a growing aggregate cluster, at which point they stick. Introduced by Witten and Sander (1981), it produces branching, tree-like snowflake structures with a fractal dimension of approximately 1.71 in 2D. DLA models physical processes including electrochemical deposition, frost crystal growth, lightning discharge paths, and mineral deposits.

What is the fractal dimension of a DLA cluster?

2D DLA clusters have a fractal dimension D ≈ 1.71. This means that if you double the radius of the cluster, the number of particles grows by a factor of 2^1.71 ≈ 3.28 rather than the factor of 4 you'd expect for a solid 2D disc (D=2). The fractal dimension can be measured by the sandbox / box-counting method: count particles N(r) within radius r; D = log N(r) / log r.

Why does DLA produce branching structures?

Branching emerges because protruding tips of the cluster intercept random walkers before they can penetrate deep into the interior — a phenomenon called screening. Once a branch forms, it acts as a lightning rod for future walkers. Small fluctuations are amplified: tips grow faster than concavities, giving the characteristic dendritic (tree-like) branching seen in snowflakes, lightning, and electrodeposits.

How does stickiness probability change the structure?

When stickiness = 1.0 (classic DLA) a particle sticks on first contact, maximising branching. Reducing stickiness (e.g. to 0.2) allows particles to bounce away and explore deeper before sticking, producing denser Eden-model-like clusters with D closer to 2. Intermediate stickiness gives fractal dimensions between 1.71 and 2, allowing continuous morphology control.

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🧪 Diffusion-Limited Aggregation (DLA) – Fractal Crystal Growth

Interactive DLA fractal growth simulator. Watch random-walking particles stick to a growing aggregate to form snowflake-like crystal fractals. 6 presets including symmetry modes. Free.

Physics & Mechanics2DModerate60 FPS

⚙ Under the hood

This simulation demonstrates the process of fractal crystal growth through diffusion-limited aggregation. Users can observe how particles accumulate to form complex, self-similar structures.

DLAFractal Growth

2D · HTML5 Canvas 2D · 60 FPS target · runs fully client-side, no install