Phase transitions and macroscopic limits in a BGK model of body-attitude coordination

Pierre Degond (1) , Antoine Diez (1) , Amic Frouvelle (2) , Sara Merino Aceituno (3, 4)

1 Department of Mathematics [Imperial College London]
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
3 Faculty of Mathematics [Vienna]
4 Department of Mathematics [Sussex]

In this article we investigate the phase transition phenomena that occur in a model of self-organisation through body-attitude coordination. Here, the body-attitude of an agent is modelled by a rotation matrix in $\R^3$ as in \cite{degondfrouvellemerino17}. The starting point of this study is a BGK equation modelling the evolution of the distribution function of the system at a kinetic level. The main novelty of this work is to show that in the spatially homogeneous case, self-organisation may appear or not depending on the local density of agents involved. We first exhibit a connection between body-orientation models and models of nematic alignment of polymers in higher dimensional space from which we deduce the complete description of the possible equilibria Then, thanks to a gradient-flow structure specific to this BGK model, we are able to prove the stability and the convergence towards the equilibria in the different regimes. We then derive the macroscopic models associated to the stable equilibria in the spirit of \cite{degondfrouvellemerino17} and \cite{degondfrouvelleliu15}.

Mots clés: Collective motion; Vicsek model; generalized collision invariant; rotation group

Domaines: Equations aux dérivées partielles [math.AP]; Physique mathématique [math-ph]

Pierre Degond, Antoine Diez, Amic Frouvelle, Sara Merino Aceituno. Phase transitions and macroscopic limits in a BGK model of body-attitude coordination. Journal of Nonlinear Science, 2020, 30, pp.2671-2736. ⟨10.1007/s00332-020-09632-x⟩. ⟨hal-02126060⟩