Vortex arrays and meso-scale turbulence of self-propelled particles
  Inspired by the Turing mechanism for pattern formation, we propose a simple self-propelled particle model with short-ranged alignment and anti-alignment at larger distances. It is able to produce orientationally ordered states, periodic vortex patterns as well as meso-scale turbulence. The latter phase resembles observations in dense bacterial suspensions. The model allows a systematic derivation and analysis of a kinetic theory as well as hydrodynamic equations for density and momentum fields. A phase diagram with regions of such pattern formation as well as spatially homogeneous orientational order and disorder is obtained from a linear stability analysis of these continuum equations. Microscopic Langevin simulations of the self-propelled particle system are in agreement with these findings. 
Robert Grossmann, Pawel Romanczuk, Markus B\"ar, Lutz Schimansky-Geier
cond-mat.soft cond-mat.stat-mech