A Boltzmann model for rod alignment and schooling fish. (14th May 2015)

• Record Type: Journal Article
• Authors: Carlen, Eric; Carvalho, Maria C; Degond, Pierre; Wennberg, Bernt
• Is Part Of: Nonlinearity. Volume 28:Number 6(2015:Jun.)
• Page Start: 1783
• Page End: 1803
• Publication Date: 2015-05-14
• Subjects: kinetic equation -- equilibrium -- swarm; 35Q20 -- 35Q70 -- 35Q82 -- 35Q92 -- 60J75 -- 60K35 -- 82C21 -- 82C22 -- 82C31 -- 92D50; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/0951-7715/28/6/1783
• Languages: English
• ISSNs: 0951-7715

Abstract

Abstract: 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.