A Boltzmann model for rod alignment and schooling fish

We consider a Boltzmann model introduced by Bertin, Droz and Grégoire as a binary interaction model of the Vicsek alignment interaction. This model considers particles lying on the circle. Pairs of particles interact by trying to reach their mid-point (on the circle) up to some noise. We study the e...

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Bibliographic Details
Published in:Nonlinearity 2015-06, Vol.28 (6), p.1783-1803
Main Authors: Carlen, Eric, Carvalho, Maria C, Degond, Pierre, Wennberg, Bernt
Format: Article
Language:English
Subjects:
Alignment
Bifurcations
equilibrium
Fish
Fourier analysis
Inverse
kinetic equation
Matematik
Mathematical models
Mathematics
Noise
Nonlinearity
swarm
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Summary:We consider a Boltzmann model introduced by Bertin, Droz and Grégoire as a binary interaction model of the Vicsek alignment interaction. This model considers particles lying on the circle. Pairs of particles interact by trying to reach their mid-point (on the circle) up to some noise. We study the equilibria of this Boltzmann model and we rigorously show the existence of a pitchfork bifurcation when a parameter measuring the inverse of the noise intensity crosses a critical threshold. The analysis is carried over rigorously when there are only finitely many non-zero Fourier modes of the noise distribution. In this case, we can show that the critical exponent of the bifurcation is exactly 1/2. In the case of an infinite number of non-zero Fourier modes, a similar behavior can be formally obtained thanks to a method relying on integer partitions first proposed by Ben-Naïm and Krapivsky.
ISSN:0951-7715
1361-6544
1361-6544
DOI:10.1088/0951-7715/28/6/1783