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

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BGKbody.pdf (701.0Kb)
Date
2019-05
Publisher city
Paris
Publisher
Cahier de recherche CEREMADE, Université Paris-Dauphine
Publishing date
05-2019
Collection title
Cahier de recherche CEREMADE, Université Paris-Dauphine
Link to item file
https://hal.archives-ouvertes.fr/hal-02126060
Dewey
Analyse
Sujet
rotation group; generalized collision invariant; Vicsek model; Collective motion
URI
https://basepub.dauphine.fr/handle/123456789/19398
Collections
  • CEREMADE : Publications
Metadata
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Author
Degond, Pierre
4553 Department of Mathematics [Imperial College London]
Diez, Antoine
4553 Department of Mathematics [Imperial College London]
Frouvelle, Amic
60 CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Merino-Aceituno, Sara
454362 Faculty of Mathematics [Vienna]
7255 Department of Mathematics [Sussex]
Type
Document de travail / Working paper
Item number of pages
52
Abstract (EN)
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 \R3 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}.

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