Hydrodynamic Fluctuations and Instabilities in Ordered Suspensions of Self-Propelled Particles

We construct the hydrodynamic equations for suspensions of self-propelled particles (SPPs) with spontaneous orientational order, and make a number of striking, testable predictions: (i) Nematic SPP suspensions are always absolutely unstable at long wavelengths. (ii) SPP suspensions support novel propagating modes at long wavelengths, coupling orientation, flow, and concentration. (iii) In a wave number regime accessible only in low Reynolds number systems such as bacteria, polar-ordered suspensions are invariably convectively unstable. (iv) The variance in the number $N$ of particles, divided by the mean $⟨N⟩$, diverges as $⟨N{⟩}^{2/3}$ in polar-ordered SPP suspensions.