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

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 $\sqrt{<(\delta N{)}^2>}=K^{\prime }<N{>}^{\varphi (d)}$ with $\varphi (d)=\frac{7}{10}+\frac{1}{5d}$ in $d\le 4$ 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 $K^{\prime }$ 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 $\frac{3\pi }{8}=67.5^{\circ }$, 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.

23 pages; Invited contribution to Journal of Chemical Physics Special Issue on Active Matter

References in corpus (1)

• Collective motion of self-propelled particles interacting without cohesion

Cited by in corpus (6)

• Natural Swarms in $3.99$ Dimensions
• Active uniaxially ordered suspensions on disordered substrates
• Hydrodynamic theory of flocking at a solid-liquid interface: long range order and giant number fluctuations
• Percolation of systems having hyperuniformity or giant number-fluctuations
• Anti-hyperuniform Critical States of Active Topological Defects
• Pseudo-giant number fluctuations and nematic order in microswimmer suspensions