Phase Transitions and Hysteresis for Self-organized Alignment Dynamics

Abstract:
In this talk I will present some rigorous description of phase transitions and symmetry breaking 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. Exponential convergence to equilibrium was proved for both sub-and supercritical cases, and algebraic convergence in the critical case. A key tool in the analysis is a new and subtle Sobolev estimate on the commutator of the gradient and conformal Laplacian operators acting on functions defined on the sphere. In the spatially inhomogeneous case, we derive the macroscopic models associated to the stable equilibria and classify their hyperbolicity according to the same function. We will also discuss the first-order phase transitions with hysteresis.

Biography:
Jian-Guo Liu earned the BS and MS at Fudan University, China, in 1982 and 1985 respectively, and the Ph.D. at University of California, Los Angeles, in 1990. He joined the Department of Mathematics at Temple University as an Assistant Professor in 1993 and moved to University of Maryland, College Park, where he became an Associate Professor in 1997, and a professor in 2001, in the Department of Mathematics and Institute for Physical Science and Technology. He joined Duke University in 2009 at the Department of Physics and Department of Mathematics. Dr. Liu’s research is in the areas of collective dynamics, decision making and self-organization in complex systems coming from biology and social sciences; scaling behavior in models of clustering and coarsening; numerical methods for incompressible viscous flow; and Multiscale Analysis and Computation. He is a fellow of AMS. He published more than 150 journal papers and gave more than 300 invited talks, colloquia and seminars.

Seminar by the NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai