干的活跃极性流体中的巨大的数字波动：与避雷针的令人震惊的类比
Giant number fluctuations in dry active polar fluids: A shocking analogy with lightning rods
摘要：
干燥的活性极性流体的流体动力学方程（即，没有动量守恒的移动群）被证明意味着巨大的数字波动。具体地说，包含平均活性粒子数量$<N>$的区域中活性粒子数量$N$的均方根波动$\sqrt {<(δN)^2>}$根据$\sqrt {<(δN)^2>} = K'<N>^{ϕ(d)}$与$ϕ(d)=\frac{7}{10}+\frac{1}{5d}$在$d\le4$空间维度上的定律进行缩放。这比在大多数平衡和非平衡系统中发现的“大数定律”标度$\sqrt {<(δN)^2>} = K\sqrt{<N>}$大得多。与大多数其他系统相比，系数$K'$还奇异地依赖于计算粒子的盒子的形状，在非常薄的盒子的极限中消失。这些涨落{ \it not}是由大的密度涨落引起的--实际上，\dry中的密度涨落一般不是特别大--而是由这些涨落之间的长程空间相关引起的。这些被证明是密切相关的，在两个空间维度上的静电势附近的一个尖锐的向上指向的导电楔的开口角${3π\over8}=67.5^\circ$，并在三维空间上的静电势附近的一个尖锐的向上指向的带电锥的开口角$37.16^\circ$。这种非常精确的预测可以通过直接测量这种密度-密度相关函数的替代盒计数实验来严格测试。
英文摘要：
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 $\sqrt {<(δN)^2>}$ of the number $N$ of active particles in a region containing a mean number of active particles $<N>$ scales according to the law $\sqrt {<(δN)^2>} = K'<N>^{ϕ(d)}$ with $ϕ(d)=\frac{7}{10}+\frac{1}{5d}$ in $d\le4$ spatial dimensions. This is much larger the "law of large numbers" scaling $\sqrt {<(δN)^2>} = K\sqrt{<N>}$ 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 ${3π\over8}=67.5^\circ$, 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.
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