Macroscopic limits and phase transition in a system of self-propelled particles

TL;DR

This paper studies a modified model of self-propelled particles with alignment interactions, revealing phase transitions from disordered to ordered states and analyzing the resulting hydrodynamic equations and their stability.

Contribution

It introduces a non-normalized force in the particle system, extending previous models and analyzing the resulting phase transition and hydrodynamic behavior.

Findings

Phase transition from disordered to aligned states at a critical density

Different convection speeds due to modified force normalization

Potential loss of hyperbolicity in the hydrodynamic model

Abstract

We investigate systems of self-propelled particles with alignment interaction. Compared to previous work, the force acting on the particles is not normalized and this modification gives rise to phase transitions from disordered states at low density to aligned states at high densities. This model is the space inhomogeneous extension of a previous work by Frouvelle and Liu in which the existence and stability of the equilibrium states were investigated. When the density is lower than a threshold value, the dynamics is described by a non-linear diffusion equation. By contrast, when the density is larger than this threshold value, the dynamics is described by a hydrodynamic model for self-alignment interactions previously derived in Degond and Motsch. However, the modified normalization of the force gives rise to different convection speeds and the resulting model may lose its hyperbolicity in some regions of the state space.