INS seminars 51: Phase transitions, hysteresis, and hyperbolicity for self-organized alignment dynamics (Jian-Guo Liu, Mar.12, 2013)

INS seminars 51

Abstract:

We provide a complete and rigorous description of phase transitions for kinetic models describing 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.

This is a joint work with Amic Frouvelle of Universite Paris-Dauphine and Pierre Degond of Universite de Toulouse.