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A Boltzmann model for rod alignment and schooling fish

E. Carlen ; M. C. Carvalho ; P. Degond ; Bernt Wennberg (Institutionen för matematiska vetenskaper)
Nonlinearity (0951-7715). Vol. 28 (2015), 6, p. 1783-1803.
[Artikel, refereegranskad vetenskaplig]

We consider a Boltzmann model introduced by Bertin, Droz and Gregoire 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-Naim and Krapivsky.

Nyckelord: kinetic equation, equilibrium, swarm

Denna post skapades 2015-06-05. Senast ändrad 2015-06-16.
CPL Pubid: 218074


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