Skip to content

Vicsek model - swarm control

Model description

We take the vicsek model described in (?): $$ x_i(t + \Delta t) = x_i(t) + v_i \Delta t$$ and we add a potential \(\Psi\):

\[X_i(t+\Delta t) = X_i(t) + v_i \Delta t - \nabla \psi (X_i(t+\Delta t), t) \Delta t\]

where \(x_i\) is the position of the ith particle and its velocity \(v_i\) is of modulus 1 and makes an angle \(\theta\) s.t. $$ \theta_i(t + \Delta t) = <\theta>_r + \eta_i$$ wehere \(<\theta>_r\) is the average orientation of particles within radius \(r\) of particle \(i\) and \(\eta_i\) is a random perturbation

vicsek with potential

Vicsek particles with a constant gaussian potential

The aim is to control the potential's location to get the particles to perform certain tasks such as target searching or tracking etc.

references

Summary
  • A novel (in 95) type of dynamics
  • investigates clustering, transport and phase transition in non-equilibrium systems
  • ONE rule (on the velocity of the particles): $$ \Delta \theta = \text{average orientation}_r + \text{noise}$$
  • main result: simple model results in rich and realistic dynamics including a kinetic phase transition from no transport to finite net transport.
  • analogy with Ising model: random fluctuations \(\approx\) temperature
  • emergence of self-ordre (low energy configuration?)