Diffusion-limited aggregation
In diffusion-limited aggregation, particles wander randomly until they touch a growing cluster and stick. This minimal rule produces an open, branching aggregate.
Exposed tips capture walkers
Outer tips are encountered more readily than screened fjords. They grow outward and shield interior regions, amplifying protrusions into branches. The feedback occurs through access probability rather than a prewritten branching template.
A random walker reaches protruding tips more often because they are accessible from many directions. Deep fjords are screened by outer branches. Once a particle sticks, it strengthens that accessibility difference. Branching is not drawn explicitly; it emerges from harmonic measure around the cluster. Small changes to launch and sticking rules affect density and anisotropy.
A stochastic fractal
Individual clusters differ, but ensembles show characteristic scaling over the simulated range. Dimension estimates describe how mass grows with radius; lattice, particle count and launch policy introduce finite and discretization effects.
Mass M within radius R grows approximately as Rᴰ across a simulated interval. D is estimated over many clusters because individual realizations vary strongly. A square lattice can reveal preferred axes; off-lattice models reduce that bias but cost more. Particle count and outer boundary limit the scale interval, so a dimension without simulation size says little.
A model family, not every mechanism
DLA captures aspects of electrodeposition, viscous fingering and aggregation where diffusion limits supply. Surface tension, fluid flow, reaction kinetics and anisotropy may require richer models. Visual resemblance alone does not select DLA.
DLA models diffusive supply followed by immediate sticking. Electrodeposition may also involve electric fields and reaction kinetics, while viscous fingering includes flow and surface tension. The minimal model isolates an important instability without replacing that physics. Fit should be judged through growth laws and statistics, not the fern-like silhouette alone.
Sources and further reading
This article summarizes the following specialist sources in original wording. Accessed and editorially reviewed 12 August 2026.
- Fractal Geometry: Mathematical Foundations and ApplicationsWiley
- The fractal geometry of lightningProceedings of the Royal Society A
Continue the idea
Lightning and dielectric breakdown
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