STUDY #13 · 2026 · IN OBSERVATION

Vicsek Model

A model-driven visual study of flocking and collective motion.

MOVING IMAGE — η TRANSITION ARC gas ⇄ flock · η 4.8 → 0.22 · hue = heading

WHAT IS THIS

Starlings, sardines, marching locusts — large groups that move as one with no conductor. The Vicsek model (1995) captures this collective motion with a single rule: each particle moves at a fixed speed and, each step, turns toward the average heading of the neighbours within a small radius, plus a little noise. That is all — yet as the noise falls the group crosses a threshold and suddenly flows together, an order-disorder transition.

This study runs the model in real time on the GPU: headings are splatted into an orientation field, and every particle reads back the neighbours in its cell — a shortcut that lets tens of thousands of particles align at once.

murmuration — an ordered flock, one hue η 0.3 · ρ 3 · v₀ 0.35 · starling · va 0.98

Motif self-propelled particles / local alignment / order-disorder transition / traveling bands

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 Lowering the noise drives a sharp order-disorder transition — the hue spread on screen reads the order parameter va directly. Denser crowds align at higher noise; at low density and low noise a dilute aligned band sweeps through the gas. Order is bound to transport: the flock is a Kuramoto that walks.

Reference T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen & O. Shochet, "Novel Type of Phase Transition in a System of Self-Driven Particles," Physical Review Letters, vol.75, 1226-1229 (1995).

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.

THE SYNCHRONIZATION TRILOGY

After Kuramoto and Ising — order that moves.

#07 Kuramoto · #10 Ising — order in place #13 Vicsek — order on the move

What aligns a phase (an oscillator's clock) · a spin (up or down) a heading — the direction of motion itself

The substrate oscillators and spins fixed in place particles that carry their order with them

What order does synchrony · magnetisation transport — the ordered crowd flows somewhere

How you read it r (coherence) · |m| (magnetisation) the spread of hue on screen is va

PARAMETERS EXPLORED

param meaning effect on the image

η noise — the disorder added to each heading every step (0-2π) the lead axis: small collapses the screen to one hue (aligned), large scatters every hue (gas). The transition sits at a threshold ηc

ρ number density N/L² denser crowds align more easily — the threshold ηc rises with density

v₀ speed — the distance moved per step near 0 a frozen texture; large and neighbours change fast, the flow sharpens into streaks

r alignment radius the reach of the interaction — the model's unit of length

Each image below records its exact parameter set.

SELECTED STILLS — 4

murmuration — an ordered flock η 0.3 · ρ 3 · v₀ 0.35 · starling · va 0.98

gas — headings everywhere η 4.0 · ρ 3 · v₀ 0.35 · starling · va 0.11

a dilute band sweeping through the gas η 0.28 · ρ 1 · v₀ 0.5 · sardine · va 0.95

sharp streaks — a fast flock η 0.4 · ρ 3 · v₀ 0.95 · locust · va 0.98

PROCESS — PARAMETER SWEEPS

The two dials that matter, swept against each other — noise η against density ρ. The ordered phase (one hue) grows out of the gas corner, and the dilute-band regime appears along the low-density edge.

the η × ρ phase map rows = ρ (density) · columns = η (noise) · colour spread = va

SIGNATURE — COLOUR IS THE ORDER PARAMETER

When the screen turns one colour, the flock has decided.

Colour encodes each particle's heading (angle → hue), the natural way to show a direction. It makes the physics visible with no instrument in between: in the ordered flock almost every particle points the same way, so the frame is nearly one hue; in the gas every hue appears together. The spread of colour on screen is, quite literally, the order parameter va.

The hero film is one slow arc of the noise dial — a gas igniting into a flock and dissolving back. You can read va off the screen the whole way.

va against η — the transition, and where the trilogy's three models differ va(η) · ηc threshold · Kuramoto / Ising / Vicsek

COLOUR = HEADING

Colour encodes each particle's heading (angle → hue). In the ordered flock the frame collapses to nearly one hue; in the gas the headings point everywhere and every hue appears at once — the colour you see IS the order parameter va.

The trailing streaks are time — each particle's recent path — which is what makes a still read as a murmuration rather than a scatter of dots.

the disordered gas — every heading, every hue, at once η 4.0 · va 0.11 · starling

Palettes (starling / sardine / locust) are artistic nods to real flocks, not measurements.

REFERENCES

T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen & O. Shochet, "Novel Type of Phase Transition in a System of Self-Driven Particles," Physical Review Letters, vol.75, 1226-1229 (1995).