Diffusion-Limited Aggregation
Particles wander randomly until they touch something solid, then stick forever. From this simple rule, fractal dendrites emerge — the same branching patterns found in snowflakes, lightning, and mineral veins.
Df ≈ 1.71 (2D DLA fractal dimension)
What is diffusion-limited aggregation?
DLA is one of the simplest models that produces fractal growth. A single seed particle sits at the center. Other particles are released far away and undergo random walks — Brownian motion, stepping randomly in every direction. When a walking particle stumbles into a neighbor of the growing aggregate, it sticks permanently. The aggregate grows outward one particle at a time, and the result is a branching, dendritic structure with no two runs producing the same shape.
The fractal dimension
A DLA cluster in two dimensions has a fractal dimension of approximately Df ≈ 1.71. This means the cluster is more than a line (D = 1) but less than a filled disk (D = 2). Measure the number of particles N within a circle of radius R from the center: you find N ∝ RDf. The cluster is full of holes and branches at every scale — a statistical self-similarity that defines fractal geometry.
Where DLA appears in nature
The same branching patterns arise whenever growth is limited by diffusion of some quantity toward an interface. Electrodeposition — metal ions diffusing through solution and plating onto a cathode — produces copper or zinc dendrites that look exactly like DLA clusters. Mineral dendrites in rock form when manganese oxide solutions seep through thin fractures. Lightning (Lichtenberg figures) branches because the electric field, which obeys the same Laplace equation as diffusion, concentrates at tips. Snowflake arms grow by vapor diffusion. Coral and river deltas branch for similar reasons — nutrients or flow arriving by diffusion, depositing at the advancing front.
Laplacian growth and the screening effect
DLA is a discrete model of Laplacian growth: the probability that a random walker arrives at a point on the boundary is proportional to the gradient of the harmonic (Laplace-equation) field at that point. Tips and promontories stick out into the field, so walkers are far more likely to land there than in the notches between branches. This is the screening effect — inner regions are screened from incoming particles by the outer branches, which is why DLA clusters are branched and sparse rather than compact. The instability that causes branching is closely related to the Mullins-Sekerka instability in solidification theory: any bump on the interface grows faster because diffusion concentrates at the bump.
The stickiness parameter
In classical DLA, a particle sticks with probability 1 on first contact. If you reduce the sticking probability, a particle that touches the aggregate may bounce off and continue its random walk. This changes the morphology: low stickiness allows particles to penetrate deeper into notches before sticking, producing denser, more compact clusters. High stickiness (the classic case) produces the most branched, open structures, because particles stick at the first tip they touch. You can explore this with the stickiness slider above.
History
DLA was introduced by Thomas Witten and Leonard Sander in 1981. Their paper showed that a simple random-walk aggregation rule generates fractal clusters, connecting statistical physics to fractal geometry. The model immediately attracted intense study because it captured the essence of pattern formation in diffusion-limited systems, and it remains one of the most studied models in non-equilibrium statistical mechanics.