Toner John
Department of Physics, Institute of Theoretical Science, University of Oregon, Eugene, Oregon 97403-5203, USA.
J Chem Phys. 2019 Apr 21;150(15):154120. doi: 10.1063/1.5081742.
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)⟩ of the number N of active particles in a region containing a mean number of active particles ⟨N⟩ scale according to the law ⟨(δN)⟩=K⟨N⟩ with ϕ(d)=710+15d in d ≤ 4 spatial dimensions. This is much larger than the "law of large numbers" scaling ⟨(δN)⟩=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.
已有研究表明[H. Chaté等人，《物理评论E》77, 046113 (2008)以及F. Ginelli，《欧洲物理杂志：专题》225, 2099 (2016)]，干性活性极性流体（即无动量守恒的移动群体）的流体动力学方程意味着巨大的数量涨落。具体而言，在包含平均活性粒子数⟨N⟩的区域中，活性粒子数N的均方根涨落⟨(δN)⟩按照⟨(δN)⟩ = K⟨N⟩的规律缩放，其中在d ≤ 4空间维度中ϕ(d)=710 + 15d 。这比大多数平衡和非平衡系统中发现的“大数定律”缩放⟨(δN)⟩ = K⟨N⟩要大得多。在本文中，证明了巨大的数量涨落还奇异地依赖于计数粒子的盒子的形状，在非常薄和非常胖的盒子的极限情况下消失。这些涨落并非源于大的密度涨落——实际上，极性有序干性活性流体中的密度涨落一般并不特别大——而是源于这些涨落之间的长程空间相关性。在二维中，这些相关性与开口角度为3π4 = 135°的尖锐向上指向的导电楔附近的静电势密切相关，在三维中，与开口角度为37.16°的尖锐向上指向的带电锥附近的静电势密切相关。这个非常精确的预测可以通过替代的盒子计数实验进行严格测试，这些实验直接测量密度 - 密度相关函数的方向依赖性以及与距离的缩放关系。
