Stability analysis for active Brownian particle models

TL;DR

This paper provides a detailed linear stability analysis of active Brownian particle models, confirming phase separation thresholds and revealing a Landau damping-like mixing mechanism in the stable regime.

Contribution

It offers the first rigorous characterization of stability regimes in scalar active matter models, including explicit thresholds and analysis of damping phenomena.

Findings

Identified explicit stability and instability thresholds depending on particle speed.

Confirmed the occurrence of motility-induced phase separation.

Discovered a Landau damping-like mixing mechanism in the stable regime.

Abstract

We carry out a comprehensive linear stability analysis of active Brownian particle systems around a constant homogeneous state. These scalar models, being important prototypes for the continuous description of active matter, are Fokker-Planck type equations in position-orientation and are known to exhibit motility-induced phase separation. We fully characterize the linear stability and instability regimes, with an explicit threshold depending on the effective speed of the particles. In this way, we rigorously confirm a conjecture on phase separation originating in the physics and applied literature. Our sharp and quantitative (in)stability results are valid both in the non-diffusive case and in the case of small angular diffusion. In the stable non-diffusive regime, we uncover a mixing mechanism reminiscent of Landau damping for the Vlasov equation, albeit with significantly weaker decay. This decay is non-integrable in time and gives rise to substantial mathematical difficulties; in particular, it prevents the use of classical perturbative arguments to treat the case of small angular diffusion.