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

## TL;DR

This paper demonstrates that dry active polar fluids exhibit giant number fluctuations due to long-range correlations, with fluctuations scaling faster than the law of large numbers and depending on the shape of the counting region.

## Contribution

It reveals a novel scaling law for number fluctuations in dry active polar fluids and links these fluctuations to electrostatic potentials near specific geometric features.

## Key findings

- Giant number fluctuations scale as N^{rac{7}{10}+rac{1}{5d}} in d dimensions.
- Fluctuation coefficient depends on the shape of the counting box, vanishing in thin boxes.
- Fluctuations are caused by long-range spatial correlations, not large density fluctuations.

## Abstract

The hydrodynamic equations of dry active polar fluids (i.e., moving flocks without momentum conservation) are shown to imply giant number fluctuations. Specifically, the rms fluctuations $\sqrt {<(\delta N)^2>}$ of the number $N$ of active particles in a region containing a mean number of active particles $<N>$ scales according to the law $\sqrt {<(\delta N)^2>} = K'<N>^{\phi(d)}$ with $\phi(d)=\frac{7}{10}+\frac{1}{5d}$ in $d\le4$ spatial dimensions. This is much larger the "law of large numbers" scaling $\sqrt {<(\delta N)^2>} = K\sqrt{<N>}$ found in most equilibrium and non-equilibrium systems. In further contrast to most other systems, the coefficient $K'$ also depends singularly on the shape of the box in which one counts the particles, vanishing in the limit of very thin boxes. These fluctuations arise {\it not} from large density fluctuations - indeed, the density fluctuations in \dry s 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 ${3\pi\over8}=67.5^\circ$, and in three dimensions to the electrostatic potential near a sharp upward pointing charged cone of opening angle $37.16^\circ$. This very precise prediction can be stringently tested by alternative box counting experiments that directly measure this density-density correlation function.