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Double milling in self-propelled swarms from kinetic theory

We present a kinetic theory for swarming systems of 
 interacting, self-propelled discrete particles. Starting from the 
 Liouville equation for the many-body problem we 
 derive a kinetic equation for the single particle probability 
 distribution function and the related macroscopic hydrodynamic equations. 
 General solutions include flocks of constant density and fixed 
 velocity and other non-trivial morphologies such as compactly 
 supported rotating mills. The kinetic theory approach leads us to the 
 identification of macroscopic structures otherwise not recognized as 
 solutions of the hydrodynamic equations, such as double mills of two 
 superimposed flows. We find the conditions allowing for the 
 existence of such solutions and compare to the case of single 
 mills.