Phase transitions, hysteresis, and hyperbolicity for self-organized alignment dynamics

Pierre Degond Université de Toulouse Amic Frouvelle Université Paris-Dauphine Jian-Guo Liu Duke University

Analysis of PDEs mathscidoc:1702.03031

Archive for Rational Mechanics and Analysis, 216, (1), 63-115, 2015.1
We provide a complete and rigorous description of phase transitions for kinetic models of self-propelled particles interacting through alignment. These models exhibit a competition between alignment and noise. Both the alignment frequency and noise intensity depend on a measure of the local alignment. We show that, in the spatially homogeneous case, the phase transition features (number and nature of equilibria, stability, convergence rate, phase diagram, hysteresis) are totally encoded in how the ratio between the alignment and noise intensities depend on the local alignment. In the spatially inhomogeneous case, we derive the macroscopic models associated to the stable equilibria and classify their hyperbolicity according to the same function.
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@inproceedings{pierre2015phase,
  title={Phase transitions, hysteresis, and hyperbolicity for self-organized alignment dynamics},
  author={Pierre Degond, Amic Frouvelle, and Jian-Guo Liu},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170207235547434185284},
  booktitle={Archive for Rational Mechanics and Analysis},
  volume={216},
  number={1},
  pages={63-115},
  year={2015},
}
Pierre Degond, Amic Frouvelle, and Jian-Guo Liu. Phase transitions, hysteresis, and hyperbolicity for self-organized alignment dynamics. 2015. Vol. 216. In Archive for Rational Mechanics and Analysis. pp.63-115. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170207235547434185284.
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