Gaussian theory for spatially distributed self-propelled particles

TL;DR

This paper introduces a Gaussian approximation method to derive continuum equations for self-propelled particles, showing it effectively captures the system's behavior and offers improvements over traditional truncation methods.

Contribution

The paper proposes a Gaussian approximation approach for deriving continuum equations, providing a better fit to simulation data than existing truncation methods.

Findings

Global polarization exhibits hysteresis with noise.

GA matches simulation results at low noise.

Spatio-temporal structures agree with simulations.

Abstract

Obtaining a reduced description with particle and momentum flux densities outgoing from the microscopic equations of motion of the particles requires approximations. The usual method, we refer to as truncation method, is to zero Fourier modes of the orientation distribution starting from a given number. Here we propose another method to derive continuum equations for interacting self-propelled particles. The derivation is based on a Gaussian approximation (GA) of the distribution of the direction of particles. First, by means of simulation of the microscopic model we justify that the distribution of individual directions fits well to a wrapped Gaussian distribution. Second, we numerically integrate the continuum equations derived in the GA in order to compare with results of simulations. We obtain that the global polarization in the GA exhibits a hysteresis in dependence on the noise intensity. It shows qualitatively the same behavior as we find in particles simulations. Moreover, both global polarizations agree perfectly for low noise intensities. The spatio-temporal structures of the GA are also in agreement with simulations. We conclude that the GA shows qualitative agreement for a wide range of noise intensities. In particular, for low noise intensities the agreement with simulations is better as other approximations, making the GA to an acceptable candidates of describing spatially distributed self-propelled particles.