Vicsek model

One motivation of the study of Active_matter by physicist is the rich phenomenology associated to this field. Collective motion and swarming are among the most studied phenomena. Among the huge number of models that can be found, the most famous is the so called Vicsek model introduced by Vicsek et al. in 1995.

Physicists have a great interest in this model as it is minimal and permits to catch a kind of universality. It consists in point like self-propelled particles that evolve at constant speed and align their velocity with their neighbours' one in presence of noise. Such a model shows collective motion at high density of particles or low noise on the alignment.

Model (mathematical description)

As this model aims at being minimal such as it puts that flocking is due to the combination of any kind of self propulsion and any kind of effective alignment, the behaviour of one particule is described by two discrete time equations of evolution. A particle $i$ is described by its position $\mathbf {r} _{i}(t)$ and the angle defining the direction of its velocity $\Theta _{i}(t)$ at time $t$. At each time steps $\Delta t$, each particle aligns with its neighbours at a distance $r$ with an incertitude due to the noise $\eta _{i}(t)$ such as :

$\Theta _{i}(t+\Delta t)=\langle \Theta _{j}\rangle _{|r_{i}-r_{j}|<r}+\eta _{i}(t)$

And moves at constant speed $v$ in the new direction :

$\mathbf {r} _{i}(t+\Delta t)=\mathbf {r} _{i}(t)+v{\begin{pmatrix}\cos(\Theta _{i}(t+\Delta t)\\sin(\Theta _{i}(t+\Delta t)\end{pmatrix}}$

The whole model is controled by two parameters: the density of particules and the amplitude of the noise on the alignment. From this two simple iteration rules diverse continuous theory have been elaborated such as the Toner Tu theory wich describes the system at the hydrodynamic level.

Phenomenology

This model shows a rich phenomenology for the phase transition from a disordered motion to a large scale ordered motion. At large noise or low density particules are in average not aligned which can be seen as a disorderd gas. At low noise and large density particules are globlally aligned and so move in the same direction (collective motion) and thus are described as an ordered liquid. the transition between those two phases is not continuous. Indeed the phase diagram exhibits a first order phase transition with a microphase separation. In the co-existence zone of the phase diagram finite size highly dense ordered bands emerge and move along their tranverse direction.

Extensions

Since it apparition in 1995 this model have been very popular in the phisicist community thus a lot fo scientists have worked on and extended it. for example one can extract several universilaty classes from simple symettry argument on the motion of the particules and their alignment.

Moreover in real systems a lot of parameters have to be taken into account in order to give more realistic description for example attraction and repulsion between agent (finite size particules), chemotaxis (bacteria), memory, non-indentical particles...

Also a simpler theory have been develloped in order to simplify the analytic approach of this model as the Active Ising model.