Pattern formation in flocking models: A hydrodynamic description

TL;DR

This paper analyzes hydrodynamic theories of collective motion in polar active matter, revealing numerous propagative solutions, their stability, and phase diagrams, connecting microscopic models to macroscopic behaviors.

Contribution

It introduces a comprehensive phenomenological framework for understanding phase and microphase separation in flocking models, including stability analysis and phase diagram construction.

Findings

Existence of infinitely many propagative solutions

Most solutions are linearly unstable, with stable ones including known phases

Coarsening leads to phase-separated states in practice

Abstract

We study in detail the hydrodynamic theories describing the transition to collective motion in polar active matter, exemplified by the Vicsek and active Ising models. Using a simple phenomenological theory, we show the existence of an infinity of propagative solutions, describing both phase and microphase separation, that we fully characterize. We also show that the same results hold specifically in the hydrodynamic equations derived in the literature for the active Ising model and for a simplified version of the Vicsek model. We then study numerically the linear stability of these solutions. We show that stable ones constitute only a small fraction of them, which however includes all existing types. We further argue that in practice, a coarsening mechanism leads towards phase-separated solutions. Finally, we construct the phase diagrams of the hydrodynamic equations proposed to qualitatively describe the Vicsek and active Ising models and connect our results to the phenomenology of the corresponding microscopic models.