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Article Dans Une Revue Nonlinearity Année : 2015

A BOLTZMANN MODEL FOR ROD ALIGNMENT AND SCHOOLING FISH

Résumé

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.
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Dates et versions

hal-02498839 , version 1 (04-03-2020)

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  • HAL Id : hal-02498839 , version 1

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Eric Carlen, Maria Carvalho, Pierre Degond, Bernt Wennberg. A BOLTZMANN MODEL FOR ROD ALIGNMENT AND SCHOOLING FISH. Nonlinearity, 2015. ⟨hal-02498839⟩
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