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 description, 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.

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 steps ${\displaystyle \Delta t}$, each agent aligns with its neighbours at a distance ${\displaystyle r}$ with an incertitude due to a noise ${\displaystyle \eta _{i}(t)}$ such as

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