Large density expansion of a hydrodynamic theory for self-propelled particles

TL;DR

This paper derives simplified hydrodynamic equations for self-propelled particles at high densities, enabling analytical study of density instabilities and stability thresholds in flocking models.

Contribution

It provides an analytical large-density expansion of transport coefficients in a kinetic theory for self-propelled particles, facilitating the study of phase stability.

Findings

Analytical expressions for instability growth rates at large densities.

Prediction of system size scaling with the square root of collision partners.

Identification of multiple scaling regimes for perturbation growth.

Abstract

Recently, an Enskog-type kinetic theory for Vicsek-type models for self-propelled particles has been proposed [T. Ihle, Phys. Rev. E 83, 030901 (2011)]. This theory is based on an exact equation for a Markov chain in phase space and is not limited to small density. Previously, the hydrodynamic equations were derived from this theory and its transport coefficients were given in terms of infinite series. Here, I show that the transport coefficients take a simple form in the large density limit. This allows me to analytically evaluate the well-known density instability of the polarly ordered phase near the flocking threshold at moderate and large densities. The growth rate of a longitudinal perturbation is calculated and several scaling regimes, including three different power laws, are identified. It is shown that at large densities, the restabilization of the ordered phase at smaller noise is analytically accessible within the range of validity of the hydrodynamic theory. Analytical predictions for the width of the unstable band, the maximum growth rate and for the wave number below which the instability occurs are given. In particular, the system size below which spatial perturbations of the homogeneous ordered state are stable is predicted to scale with $\sqrt{M}$ where $M$ is the average number of collision partners. The typical time scale until the instability becomes visible is calculated and is proportional to M.