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. Within the huge number of models that have been developed to catch such behavior from a microscopic descripton, the most famous is the so called Vicsek model introduced by Tamás 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.

File:Phase diagram Vicsek.png

The phase diagram of the Vicsek model in the (density, noise) plane with the three different phases

Model (mathematical description)

As this model aims at being minimal, it assumes that flocking is due to the combination of any kind of self propulsion and of effective alignment.

An individual $i$ is described by its position $\mathbf {r} _{i}(t)$ and the angle defining the direction of its velocity $\Theta _{i}(t)$ at time $t$. The discrete time evolution of one particle is set by two equations: At each time steps $\Delta t$, each agent aligns with its neighbours at a distance $r$ with an incertitude due to a 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 these two simple iteration rules diverse continuous theories have been elaborated such as the Toner Tu theory wich describes the system at the hydrodynamic level.

Phenomenology

File:Band vicsek lg.jpg

The smectic bands of the Vicsek model (picture of the density). The red arrows indicate the direction of motion.

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 particles are in average not aligned, and they can be described as a disorderd gas. At low noise and large density, particles are globlally aligned and move in the same direction (collective motion). This state is interpreted as an ordered liquid. The transition between those two phases is not continuous, indeed the phase diagram of the system exhibits a first order phase transition with a microphase separation. In the co-existence region finite size liquid bands emerge in a gas environment and move along their tranverse direction. This spontaneous organization of particles epitomize collective motion.

Extensions

Since it apparition in 1995 this model has been very popular in the physicist community, thus a lot of scientists have worked on and extended it. For example one can extract several universilaty classes from simple symmetry arguments on the motion of the particules and their alignment.

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

Also a simpler theory has been develloped in order to facilitate the analytic approach of this model and is known as the Active Ising model.