BIBLIOS

Document type
Journal articles

Document subtype
Full paper

Title
A Boltzmann model for rod alignment and schooling fish

Participants in the publication
Eric A. Carlen (Author)
RUTGERS UNIVERSITY
Maria C. Carvalho (Author)
Dep. Matemática
CMAFcIO
Pierre Degond (Author)
IMPERIAL COLLEGE LONDON
Bernt Wennberg (Author)
CHALMERS TEKNISKA HOEGSKOLA AB

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.

Editor(s)
IOP Publishing Ltd & London Mathematical Society


Where published
NONLINEARITY

Publication Identifiers
ISSN - 0951-7715



Document Identifiers
URL - http://iopscience.iop.org/article/10.1088/0951-7715/28/6/1783