Diffusion in a continuum model of self-propelled particles with alignment interaction

TL;DR

This paper derives first-order scale corrections to a hydrodynamic model of self-propelled particles with alignment, incorporating diffusion and complex derivative terms, using an advanced Chapman-Enskog approach.

Contribution

It provides the $O()$ corrected hydrodynamic model for flocking agents, including diffusion terms, based on a detailed Chapman-Enskog derivation.

Findings

Derived $O()$ corrections with diffusion terms.

Identified quadratic derivative terms in the model.

Addressed complexities due to non-isotropy and lack of momentum conservation.

Abstract

In this paper, we provide the $O(\epsilon)$ corrections to the hydrodynamic model derived by Degond and Motsch from a kinetic version of the model by Vicsek & coauthors describing flocking biological agents. The parameter $\epsilon$ stands for the ratio of the microscopic to the macroscopic scales. The $O(\epsilon)$ corrected model involves diffusion terms in both the mass and velocity equations as well as terms which are quadratic functions of the first order derivatives of the density and velocity. The derivation method is based on the standard Chapman-Enskog theory, but is significantly more complex than usual due to both the non-isotropy of the fluid and the lack of momentum conservation.