STUDY #09 · 2026 · IN OBSERVATION

Diffusion-Limited Aggregation

A model-driven visual study of dendritic growth by diffusion.

MOVING IMAGE — GROWTH ARC η 1 · point seed · arrival-time colour

WHAT IS THIS

Diffusion-limited aggregation (DLA) is how a branched, fractal cluster grows by chance. Particles arrive one at a time from far away, wander at random like pollen in water, and freeze in place the instant they touch the cluster. Growth is limited by diffusion: a wandering particle is far likelier to reach an exposed tip than to thread its way into a narrow gap, so tips screen the interior and the pattern amplifies its own branching into a self-similar dendrite (fractal dimension ≈ 1.71 in the plane).

This study grows the model in real time on the GPU using its field cousin — the dielectric-breakdown / η-model — solving for a potential around the cluster and extending the tips with a probability set by the local field. A single exponent η tunes the whole morphology, from a compact mass to an open fern to sparse, lightning-like arms.

an open dendrite, its front still reaching — the frost palette η 1 · point seed · arrival-time colour

Motif diffusion-limited aggregation / dendrites / fractal growth / dielectric breakdown / arrival-time

Method A small simulator was generated and modified with AI assistance, then ported to a real-time GPU (GLSL) renderer. The visual output was selected through parameter exploration.

Observation Particles arriving one at a time and sticking on first contact build a branched, self-similar dendrite: tips screen the interior (diffusion-limited), so growth amplifies its own branching (fractal dimension ≈ 1.71). A single exponent η sweeps the morphology from compact to open to sparse, and each frozen point keeps the time it was added, so colour is the growth's own history.

Tools Python / NumPy / SciPy / three.js / React / GLSL / ffmpeg / AI coding assistant

Year 2026

This is not a scientific simulation result, but a visual interpretation of the phenomenon.

A NEW CHAPTER

The one study you watch build, not settle.

Studies #01–08 — fields Study #09 — growth

How it moves a field advanced by an equation, evolving in place irreversible aggregation — one stuck point at a time

The image settles, oscillates, or coarsens toward a state only ever grows — a point, once frozen, is never undone

Colour a field quantity — concentration, phase, temperature arrival time — the order in which each point was added

PARAMETERS EXPLORED

param meaning effect on the image

η the growth exponent (the master knob) growth probability ∝ field^η: η = 1 is DLA (an open fern); small η goes compact (Eden); large η goes sparse and lightning-like

Ps the sticking probability at the surface 1 is maximally sparse (tip-dominated); smaller lets particles crawl the surface into bays, filling the interior

m noise reduction — hits needed before a site freezes 1 gives stringy ferns; larger reveals the lattice anisotropy — thick, four-fold crystalline dendrites (soot → snowflake)

drift an external field (gravity in electrodeposition, airflow in frost) the cluster leans, its branches combing into alignment — frost on a windswept pane

seed the geometry of the nucleus point → a radial dendrite; ring → an outward corona; line → a forest of frost rising off a substrate

arrival time the generation each point was frozen in the colour itself — an old dim core out to a bright, still-reaching front

Each image below records its exact parameter set.

THE MATHEMATICS the model behind the images

On a lattice, random walkers stick where they first touch the cluster. The engine runs its continuous-field cousin — the dielectric-breakdown model — solving for a potential and growing the tips by the local field.

\nabla^2 \phi = 0 \qquad (\phi = 0 \ \text{on the cluster},\ \ \phi \to 1 \ \text{far away})

The potential around the cluster satisfies Laplace's equation — the same field a diffusing particle's arrival probability obeys.

p_i \;\propto\; \lvert \nabla \phi_i \rvert^{\,\eta}

Each interface site grows with a probability set by the local field gradient (the harmonic measure) raised to η. At η = 1 this is exactly DLA.

Inspired by diffusion-limited aggregation and the dielectric-breakdown / η-model — a visual interpretation, not an exact reproduction. Fractal dimension D ≈ 1.71 in the plane.

SELECTED STILLS — 3

frost — water vapour freezing on cold glass navy → steel teal → ice-white edge

ember — electrodeposition and Lichtenberg discharge dark → magenta → gold tip

silver — manganese-oxide dendrites in agate slate → pale metallic sheen

COLOUR = ARRIVAL TIME

DLA is the shape shared by many real diffusion-limited growths, and each palette is grounded in one of them: frost — water vapour freezing on cold glass; ember — electrodeposition and the Lichtenberg tracks of an electric discharge; silver — the manganese-oxide "tree" dendrites in agate and native-metal crystals.

The colour within each is not arbitrary: it maps the arrival time, the order in which each point was added, so the dim core is the oldest material and the bright edge is where the crystal is still reaching — the growth's chronology made visible, like topological tree rings.

The colours are an artistic mapping of the model's own growth order, not measured quantities.

Palettes frost / ember / silver — hue = arrival time (growth order) · the glowing edge is the still-reaching front.

REFERENCES

T. A. Witten, Jr. & L. M. Sander, "Diffusion-Limited Aggregation, a Kinetic Critical Phenomenon," Physical Review Letters, vol.47, 1400-1403 (1981).
L. Niemeyer, L. Pietronero & H. J. Wiesmann, "Fractal Dimension of Dielectric Breakdown," Physical Review Letters, vol.52, 1033-1036 (1984).