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We consider a Boltzmann model introduced by Bertin, Droz and Greegoire as abinary interaction model of the Vicsek alignment interaction. This modelconsiders particles lying on the circle. Pairs of particles interact by tryingto reach their mid-point (on the circle) up to some noise. We study theequilibria of this Boltzmann model and we rigorously show the existence of apitchfork bifurcation when a parameter measuring the inverse of the noiseintensity crosses a critical threshold. The analysis is carried over rigorouslywhen there are only finitely many non-zero Fourier modes of the noisedistribution. In this case, we can show that the critical exponent of thebifurcation is exactly 1/2. In the case of an infinite number of non-zeroFourier modes, a similar behavior can be formally obtained thanks to a methodrelying on integer partitions first proposed by Ben-Naim and Krapivsky.

A Boltzmann model for rod alignment and schooling fish