Self-propelled particle systems and the steady states in continuum limit

Abstract

Complex behaviours of animal swarming such as fish schools and bird flocks exhibit macroscopic patterns from interactions between individuals. These phenomena can be modelled both by agent-based particle systems and their continuum limiting PDEs, providing a mathematical framework frequently used in crowd dynamics, opinion formation and cell biology.

In this talk, I will first discuss how complicated behaviours could arise from a few simple rules, and then focus on a self-propelled particle systems for concurrent moving flocks and rotating mills, with simple attractive-repulsive interactions.

The continuum (or mean field) limit of the particle system is then studied, showing that the concept of H-stable potentials originated from statistical mechanics for discrete particles is equivalent to the existence of stationary patterns for the continuum integro-differential equation for a special Quasi-Morse potential. This is a joint work with José Carrillo and Stephan Martin.