Vicsek flock

Simple rules can lead to complex collective behavior. The Vicsek model illustrates this vividly: each particle is only concerned with trying to move in the same direction as its nearby peers, yet the behavior of the flock seems complex and rich.

Boundary

Particles(10–2000)

Noise (temperature)0.15

Alignment strength0.70

Interaction radius0.30

Speed1.60

Pointer

Order parameter φ = 0.00

Drag your pointer across the flock to push it around, or switch the pointer to attract and gather it instead. The order parameter under the canvas measures how aligned the flock is: turn the noise up past roughly a third and watch it collapse.

The setup

$N$ particles move at constant speed $v_0$ . At each step, a particle takes the mean heading of every neighbour within a radius $R$ and adds a random kick $\xi_i$ drawn uniformly from $[-\pi, \pi]$ , scaled by the noise $\eta$ :

\begin{aligned} \theta_i(t+\Delta t) &= \big\langle \theta_j \big\rangle_{|r_i - r_j| < R} + \eta\,\xi_i ,\\ r_i(t+\Delta t) &= r_i(t) + v_0\,\hat{e}(\theta_i)\,\Delta t \end{aligned}

Order is measured by $\varphi = \big|\langle \hat{e}(\theta_i)\rangle\big|$ , the length of the mean unit velocity: 1 for a flock moving as one, $\sim 1/\sqrt{N}$ for independent headings.