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

Degond, Pierre; Frouvelle, Amic; Liu, Jian-Guo (2015), Phase transitions, hysteresis, and hyperbolicity for self-organized alignment dynamics, Archive for Rational Mechanics and Analysis, 216, 1, p. 63-115. http://dx.doi.org/10.1007/s00205-014-0800-7

Type
Article accepté pour publication ou publié
Date
2015
Journal name
Archive for Rational Mechanics and Analysis
Volume
216
Number
1
Publisher
Springer
Pages
63-115
Publication identifier
Metadata
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Author(s)
Degond, Pierre
Frouvelle, Amic cc
Liu, Jian-Guo
Abstract (EN)
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.
Subjects / Keywords
von Mises-Fisher distribution