ChapmanEnskog expansion for the Vicsek model of selfpropelled 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 Nparticle probability distribution and, after making the meanfield assumption of molecular chaos, leads to a multiparticle Enskogtype equation. This equation is treated by a nonstandard ChapmanEnskog expansion to extract the macroscopic behavior. The expansion includes terms up to third order in a formal expansion parameter ɛ, and involves a fast time scale. A selfconsistent closure of the moment equations is presented that leads to a continuity equation for the particle density and a NavierStokeslike 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 ChapmanEnskog approach is checked by an independent calculation of the shear viscosity using a GreenKubo relation.
 Publication:

Journal of Statistical Mechanics: Theory and Experiment
 Pub Date:
 August 2016
 DOI:
 10.1088/17425468/2016/08/083205
 arXiv:
 arXiv:1605.03953
 Bibcode:
 2016JSMTE..08.3205I
 Keywords:

 Condensed Matter  Statistical Mechanics;
 Condensed Matter  Soft Condensed Matter;
 Physics  Biological Physics;
 Physics  Fluid Dynamics
 EPrint:
 doi:10.1088/17425468/2016/08/083205