Chapman–Enskog expansion for the Vicsek model of self-propelled particles

Abstract

Using the standard Vicsek model, I show how the macroscopic transport equations can be systematically derived from microscopic collision rules. The approach starts with the exact evolution equation for the N-particle probability distribution and, after making the mean-field assumption of molecular chaos, leads to a multi-particle Enskog-type equation. This equation is treated by a non-standard Chapman–Enskog expansion to extract the macroscopic behavior. The expansion includes terms up to third order in a formal expansion parameter ϵ, and involves a fast time scale. A self-consistent closure of the moment equations is presented that leads to a continuity equation for the particle density and a Navier–Stokes-like equation for the momentum density. Expressions for all transport coefficients in these macroscopic equations are given explicitly in terms of microscopic parameters of the model. The transport coefficients depend on specific angular integrals which are evaluated asymptotically in the limit of infinitely many collision partners, using an analogy to a random walk. The consistency of the Chapman–Enskog approach is checked by an independent calculation of the shear viscosity using a Green–Kubo relation.