The Vicsek Model Lab: Exploring Flocking Behavior in a 2D Torus

What is the Vicsek Model?

The Vicsek model is a mathematical framework used to simulate collective motion in biological systems such as bird flocks or fish schools. Each particle moves according to simple rules: align with neighbors, move at constant speed, and adjust direction based on local interactions.

Particles are placed on a 2D torus (a flat surface where edges wrap around), ensuring that the system is periodic.

Order-Disorder Transition in Flocking

As noise or perturbations increase, the flocking behavior transitions from an ordered state to a disordered one. This transition can be observed through changes in the order parameter, which quantifies the degree of alignment among particles.

The model demonstrates how even simple interactions between individuals can lead to complex emergent behaviors.

Neighbourhood and Probes

In the Vicsek model lab, a probe neighbourhood is used to define the range within which each particle interacts with its neighbors. This parameter significantly influences the flocking behavior.

The live order parameter provides real-time feedback on the degree of alignment in the system.

Noise-Driven Transition and Exact Results

By varying the noise level, one can observe how it drives the transition from an ordered to a disordered state. The model is compared against exact results for all-to-all interactions, represented by the function sin(eta/2)/(eta/2), where eta is the angle between particle velocities.

The random-walk floor represents the baseline behavior expected in the absence of any alignment tendency.

Frequently asked questions

Try it live

Everything above runs in your browser — open Vicsek Model Lab: Noise-Driven Flocking Order-Disorder Transition — 2D Torus & Order Curve and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.