Giant number fluctuations in dry active polar fluids: A shocking analogy with lightning rods

It has been shown [H. Chaté et al., Phys. Rev. E 77, 046113 (2008) and F. Ginelli, Eur. Phys. J.: Spec. Top. 225, 2099 (2016)] that the hydrodynamic equations of dry active polar fluids (i.e., moving flocks without momentum conservation) imply giant number fluctuations. Specifically, the rms fluctuations ⟨(δN)²⟩ of the number N of active particles in a region containing a mean number of active particles ⟨N⟩ scale according to the law ⟨(δN)²⟩=K^'⟨N⟩^ϕ(d) with ϕ(d)=710+15d in d ≤ 4 spatial dimensions. This is much larger than the "law of large numbers" scaling ⟨(δN)²⟩=K⟨N⟩ found in most equilibrium and nonequilibrium systems. In this paper, it is demonstrated that giant number fluctuations also depend singularly on the shape of the box in which one counts the particles, vanishing in the limit of very thin and very fat boxes. These fluctuations arise not from large density fluctuations-indeed, the density fluctuations in polar ordered dry active fluids are not in general particularly large-but from long ranged spatial correlations between those fluctuations. These are shown to be closely related in two spatial dimensions to the electrostatic potential near a sharp upward pointing conducting wedge of opening angle 3π4=135° and in three dimensions to the electrostatic potential near a sharp upward pointing charged cone of opening angle 37.16°. This very precise prediction can be stringently tested by alternative box counting experiments that directly measure the direction dependence, as well as the scaling with distance, of the density-density correlation function.