The Vicsek model is a mathematical model used to describe active matter. One motivation of the study of active matter by physicists 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 model introduced by Tamás Vicsek et al. in 1995. Physicists have a great interest in this model as it is minimal and describes 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, it assumes that flocking is due to the combination of any kind of self propulsion and of effective alignment. Since the speed of each particle is a constant, the net momentum of the system is not conserved during collisions. An individual i {\displaystyle i} is described by its position r i ( t ) {\displaystyle \mathbf {r} _{i}(t)} and the angle defining the direction of its velocity Θ i ( t ) {\displaystyle \Theta _{i}(t)} at time t {\displaystyle t} . The discrete time evolution of one particle is set by two equations:
At each time step Δ t {\displaystyle \Delta t} , each agent aligns with its neighbours within a given distance r {\displaystyle r} with an uncertainty due to a noise η i ( t ) {\displaystyle \eta _{i}(t)} :
Θ i ( t + Δ t ) = ⟨ Θ j ⟩ | r i − r j | < r + η i ( t ) {\displaystyle \Theta _{i}(t+\Delta t)=\langle \Theta _{j}\rangle _{|r_{i}-r_{j}|<r}+\eta _{i}(t)}
The particle then moves at constant speed v {\displaystyle v} in the new direction:
r i ( t + Δ t ) = r i ( t ) + v Δ t ( cos Θ i ( t ) sin Θ i ( t ) ) {\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}}}
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