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The Vicsek Model

Collective Motion

Coordinated behaviour is extremely common in animals. One such example is the phenomenon of collective motion. It is the spontaneous emergence of ordered movement in a system consisting of many self-propelled agents, for example flocking in birds, swarming in insects, schooling fishes, etc.

Vicsek model is a mathematical model describing the emergent properties in self-propelled particles. The central idea is that every particle aligns itself in the average direction of motion of its neighbours.

The Model

A box of size L was taken with periodic boundary conditions. We consider point particles moving on the plance. The particles have a certain radius of interaction beyond which they cannot percieve their surroundings. Positions were updated at an interval of $\Delta t=1$

The particle positions are randomly initialized within the L $\times$ L box. Each of the particles are given a fixed speed $v$ at a random direction $\theta$

The positions are updated as follows

\[ x_i (t+1) = x_i(t) + v_i (t)\Delta t \]

The direction of motion of the particles after every time step is given by,

\[ \theta (t+1) = \langle\theta(t)\rangle_r + \Delta \theta \]

$\langle\theta(t)\rangle_r$ is the average direction of the velocities of particles (including particle i) being within a circle of radius r surrounding the given particle.

In our case $\Delta \theta$ was taken to be a random number chosen with uniform probability from the interval $[\ \eta/2, \eta/2]$

Thus we have 3 free parameters for a system of given size L:

  • $\eta$
  • $\rho\ (=\frac{N}{L^2})$
  • $v$.

Simulation

Color of a particle corresponds to its direction of motion.

References

  1. Vicsek, Tamás; Czirók, András; Ben-Jacob, Eshel; Cohen, Inon; Shochet, Ofer (1995-08-07). "Novel Type of Phase Transition in a System of Self-Driven Particles".