Hyperuniformity classes of quasiperiodic tilings via diffusion spreadability

Hyperuniform point patterns can be classified by the hyperuniformity scaling exponent $\alpha >0$, that characterizes the power-law scaling behavior of the structure factor $S\left(k\right)$ as a function of wave number $k\equiv \left|k\right|$ in the vicinity of the origin, e.g., $S\left(k\right)\sim {\left|k\right|}^{\alpha }$ in cases where $S\left(k\right)$ varies continuously with $k$ as $k\to 0$. In this paper, we show that the spreadability is an effective method for determining $\alpha$ for quasiperiodic systems where $S\left(k\right)$ is discontinuous and consists of a dense set of Bragg peaks. It has been shown in [Phys. Rev. E 104, 054102 (2021)] that, for media with finite $\alpha$, the long-time behavior of the excess spreadability $S\left(\infty \right)-S\left(t\right)$ can be fit to a power law of the form $t^{-(d-\alpha )/2}$, where $d$ is the space dimension, to accurately extract $\alpha$ for the continuous case. 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 ${\tilde{\chi }}_{{}_V}\left(k\right)$ 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 $\alpha$ for the one-dimensional (1D) limit-periodic period-doubling chain ($\alpha =1$) and the 1D quasicrystalline Fibonacci chain ($\alpha =3$) to within $0.02\%$ of the analytically known exact results. Moreover, we obtain a value of $\alpha =5.97\pm 0.06$ for the two-dimensional Penrose tiling and present plausible theoretical arguments strongly suggesting that $\alpha$ is exactly equal to six. We also show that, due to the self-similarity of the structures examined here, one can truncate the small-$k$ region of the scattering information used to compute the spreadability and obtain an accurate value of $\alpha$, with a small deviation from the untruncated case that decreases as the system size increases. This strongly suggests that one can obtain a good estimate of $\alpha$ for an infinite self-similar quasicrystal from a modestly sized finite sample. The methods described here offer a simple and general procedure to characterize accurately 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 ${\tilde{\chi }}_{{}_V}\left(k\right)$, can be used to estimate their physical properties, such as their effective dynamic dielectric constants, effective dynamic elastic constants, and fluid permeabilities.