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

TL;DR

This paper rigorously analyzes phase transitions in kinetic models of self-propelled particles, revealing how local alignment and noise influence equilibria, stability, and hyperbolicity in both homogeneous and inhomogeneous cases.

Contribution

It provides a complete theoretical description of phase transitions, hysteresis, and hyperbolicity for alignment-based particle models, linking microscopic interactions to macroscopic behavior.

Findings

Phase transition characteristics depend on the ratio of alignment to noise.

Equilibria and stability are fully determined by local alignment measures.

Macroscopic models' hyperbolicity classification aligns with microscopic stability analysis.

Abstract

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.