通过扩散可扩展性研究准周期拼砌的超均匀性类别
Hyperuniformity Classes of Quasiperiodic Tilings via Diffusion Spreadability
AI总结：
本文提出利用扩散可扩展性有效确定准周期系统的超均匀性标度指数$α$，通过将点图案映射为两相介质并拟合过剩可扩展性的长时间行为，准确提取了1D倍周期链和Fibonacci链的$α$值，并首次计算出2D Penrose拼砌的$α$值。
摘要：
超均匀点图案可以通过超均匀性标度指数 $α> 0$ 进行分类，该指数表征了结构因子 $S(\mathbf{k})$ 作为波数 $k\equiv|\mathbf{k}|$ 的函数在原点附近的幂律标度行为，例如，在 $S(\mathbf{k})$ 随 $k$ 连续变化且 $k\rightarrow0$ 的情况下，$S(\mathbf{k})\sim|\mathbf{k}|^α$。在本文中，我们表明对于 $S(\mathbf{k})$ 不连续且由稠密的 Bragg 峰组成的准周期系统，spreadability 是确定 $α$ 的有效方法。我们首先将准周期和极限周期点图案转化为两相介质，方法是将它们映射到相同且不重叠的圆盘堆积上，其中圆盘内部的空间代表一相，圆盘外部的空间代表第二相。然后，我们计算这些堆积的谱密度，最后计算并拟合它们过剩 spreadability 的长时间行为。具体而言，我们表明过剩 spreadability 可用于准确提取 1D 极限周期倍周期链和 1D 准晶 Fibonacci 链的 $α$，精度在解析已知精确结果的 $0.02\%$ 以内。此外，对于先前未曾计算过的 2D Penrose 拼砌，我们获得了 $α= 5.97\pm0.06$ 的值。我们还表明，可以截断用于计算 spreadability 的散射信息的小 $k$ 区域，并且仍然能获得准确的 $α$ 值。这里描述的方法提供了一种简单的方法来表征在任何空间维度上自相似的准晶和极限周期介质中存在的大尺度平移序。此外，从编码在谱密度中的这些两相介质提取的散射信息可用于估计它们的物理性质。
英文摘要：
Hyperuniform point patterns can be classified by the hyperuniformity scaling exponent $α> 0$, that characterizes the power-law scaling behavior of the structure factor $S(\mathbf{k})$ as a function of wavenumber $k\equiv|\mathbf{k}|$ in the vicinity of the origin, e.g., $S(\mathbf{k})\sim|\mathbf{k}|^α$ in cases where $S(\mathbf{k})$ varies continuously with $k$ as $k\rightarrow0$. In this paper, we show that the spreadability is an effective method for determining $α$ for quasiperiodic systems where $S(\mathbf{k})$ is discontinuous and consists of a dense set of Bragg peaks. We first transform quasiperiodic and limit-periodic point patterns into two-phase media by mapping them onto packings of identical nonoverlapping disks, where space interior to the disks represents one phase and the space in exterior to them represents the second phase. We then compute the spectral density of the packings, and finally compute and fit the long-time behavior of their excess spreadabilities. Specifically, we show that the excess spreadability can be used to accurately extract $α$ for the 1D limit-periodic period doubling chain and the 1D quasicrystalline Fibonacci chain to within $0.02\%$ of the analytically known exact results. Moreover, we obtain a value of $α= 5.97\pm0.06$ for the 2D Penrose tiling, which had not been computed previously. We also show that one can truncate the small-$k$ region of the scattering information used to compute the spreadability and still obtain an accurate value of $α$. The methods described here offer a simple way to characterize the large-scale translational order present in quasicrystalline and limit-periodic media in any space dimension that are self-similar. Moreover, the scattering information extracted from these two-phase media encoded in the spectral density can be used to estimate their physical properties. (abridged)
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