Vicsek model

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 ${\displaystyle i}$ is described by its position ${\displaystyle \mathbf {r} _{i}(t)}$ and the angle defining the direction of its velocity ${\displaystyle \Theta _{i}(t)}$ at time ${\displaystyle t}$. The discrete time evolution of one particle is set by two equations: at each time step ${\displaystyle \Delta t}$, each agent aligns with its neighbours at a distance ${\displaystyle r}$ with an uncertainty due to a noise ${\displaystyle \eta _{i}(t)}$ such as

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

and moves at constant speed ${\displaystyle v}$ in the new direction:

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

The whole model is controlled by two parameters: the density of particles and the amplitude of the noise on the alignment. From these two simple iteration rules, various continuous theories[2] have been elaborated such as the Toner Tu theory[3] which describes the system at the hydrodynamic level.

This model shows a phase transition[4] from a disordered motion to large-scale ordered motion. At large noise or low density, particles are on average not aligned, and they can be described as a disordered gas. At low noise and large density, particles are globally aligned and move in the same direction (collective motion). This state is interpreted as an ordered liquid. The transition between these 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[5] emerge in a gas environment and move along their transverse direction. This spontaneous organization of particles epitomizes collective motion.

Since its appearance in 1995 this model has been very popular within the physics community; many scientists have worked on and extended it. For example, one can extract several universality classes from simple symmetry arguments concerning the motion of the particles and their alignment.[6]

Moreover, in real systems, many parameters can be included in order to give a more realistic description, for example attraction and repulsion between agents (finite-size particles), chemotaxis (biological systems), memory, non-identical particles, the surrounding liquid...

A simpler theory, the Active Ising model[7], has been developed to facilitate the analysis of the Vicsek model.