A Boltzmann model for rod alignment and schooling fish

• Published in: Nonlinearity Vol. 28; no. 6; pp. 1783 - 1803
• Main Authors: Carlen, Eric, Carvalho, Maria C, Degond, Pierre, Wennberg, Bernt
• Format: Journal Article
• Language: English
• Published: IOP Publishing 01.06.2015
• Subjects: Alignment; Bifurcations; equilibrium; Fish; Fourier analysis; Inverse; kinetic equation; Matematik; Mathematical models; Mathematical sciences; Mathematics; Noise; Nonlinearity; swarm; equilibria; kinetic equation; mid-point rule; pitchfork bifurca- tion; binary interaction; swarms AMS Subject Classification: 35Q20; integer partition
• ISSN: 0951-7715; 1361-6544; 1361-6544
• DOI: 10.1088/0951-7715/28/6/1783

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.