Vicsek model
vicsek modelOne 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 socalled 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 selfpropelled 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
Contents
 1 Model mathematical description
 2 Phenomenology
 3 Extensions
 4 References
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 r i t _t and the angle defining the direction of its velocity Θ i t t at time t The discrete time evolution of one particle is set by two equations: At each time steps Δ t , each agent aligns with its neighbours at a distance r with an incertitude due to a noise η i t t such as
Θ i t + Δ t = ⟨ Θ j ⟩  r i − r j  < r + η i t t+\Delta t=\langle \Theta _\rangle _r_<r+\eta _t
And moves at constant speed v in the new direction :
r i t + Δ t = r i t + v Δ t cos Θ i t sin Θ i t _t+\Delta t=\mathbf _t+v\Delta t\cos \Theta _t\\\sin \Theta _t\end
The whole model is controlled 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 which describes the system at the hydrodynamic level
Phenomenology
This model shows a 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 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 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 coexistence region finite size liquid bands emerge in a gas environment and move along their transverse direction This spontaneous organization of particles epitomize collective motion
Extensions
Since its appearance 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 universality classes from simple symmetry arguments on the motion of the particles 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 particles, chemotaxis biological systems, memory, nonidentical particles, the surrounding liquid
Also a simpler theory has been developed in order to facilitate the analytic approach of this model and is known as the Active Ising model
References
 ^ Vicsek, Tamás; Czirók, András; BenJacob, Eshel; Cohen, Inon; Shochet, Ofer 19950807 "Novel Type of Phase Transition in a System of SelfDriven Particles" Physical Review Letters 75 6: 1226–1229 doi:101103/PhysRevLett751226 PMID 10060237
 ^ Bertin, Eric; Droz, Michel; Grégoire, Guillaume 20060802 "Boltzmann and hydrodynamic description for selfpropelled particles" Physical Review E 74 2: 022101 doi:101103/PhysRevE74022101
 ^ Toner, John; Tu, Yuhai 19951204 "LongRange Order in a TwoDimensional Dynamical $\mathrm$ Model: How Birds Fly Together" Physical Review Letters 75 23: 4326–4329 Bibcode:1995PhRvL754326T doi:101103/PhysRevLett754326
 ^ Grégoire, Guillaume; Chaté, Hugues 20040115 "Onset of Collective and Cohesive Motion" Physical Review Letters 92 2: 025702 Bibcode:2004PhRvL92b5702G doi:101103/PhysRevLett92025702
 ^ Solon, Alexandre P; Chaté, Hugues; Tailleur, Julien 20150212 "From Phase to Microphase Separation in Flocking Models: The Essential Role of Nonequilibrium Fluctuations" Physical Review Letters 114 6: 068101 Bibcode:2015PhRvL114f8101S doi:101103/PhysRevLett114068101
 ^ Chaté, H; Ginelli, F; Grégoire, G; Peruani, F; Raynaud, F 20080711 "Modeling collective motion: variations on the Vicsek model" The European Physical Journal B 64 34: 451–456 doi:101140/epjb/e2008002759 ISSN 14346028
 ^ Solon, A P; Tailleur, J 20130813 "Revisiting the Flocking Transition Using Active Spins" Physical Review Letters 111 7: 078101 Bibcode:2013PhRvL111g8101S doi:101103/PhysRevLett111078101
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29.10.2014
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