A Boltzmann model for rod alignment and schooling fish

Eric Carlen, Maria C. Carvalho, Pierre Degond, Bernt Wennberg

Research output: Contribution to journalArticlepeer-review

12 Scopus citations


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.

Original languageEnglish (US)
Article number1783
Pages (from-to)1783-1803
Number of pages21
Issue number6
StatePublished - Jun 1 2015

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Physics and Astronomy(all)
  • Applied Mathematics


  • equilibrium
  • kinetic equation
  • swarm

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