Characterizing pixel and point patterns with a hyperuniformity disorder length

We introduce the concept of a “hyperuniformity disorder length” $h$ that controls the variance of volume fraction fluctuations for randomly placed windows of fixed size. In particular, fluctuations are determined by the average number of particles within a distance $h$ from the boundary of the window. We first compute special expectations and bounds in $d$ dimensions, and then illustrate the range of behavior of $h$ versus window size $L$ by analyzing several different types of simulated two-dimensional pixel patterns—where particle positions are stored as a binary digital image in which pixels have value zero if empty and one if they contain a particle. The first are random binomial patterns, where pixels are randomly flipped from zero to one with probability equal to area fraction. These have long-ranged density fluctuations, and simulations confirm the exact result $h=L/2$. Next we consider vacancy patterns, where a fraction $f$ of particles on a lattice are randomly removed. These also display long-range density fluctuations, but with $h=(L/2)(f/d)$ for small $f$, and $h=L/2$ for $f\to 1$. And finally, for a hyperuniform system with no long-range density fluctuations, we consider “Einstein patterns,” where each particle is independently displaced from a lattice site by a Gaussian-distributed amount. For these, at large $L,h$ approaches a constant equal to about half the root-mean-square displacement in each dimension. Then we turn to gray-scale pixel patterns that represent simulated arrangements of polydisperse particles, where the volume of a particle is encoded in the value of its central pixel. And we discuss the continuum limit of point patterns, where pixel size vanishes. In general, we thus propose to quantify particle configurations not just by the scaling of the density fluctuation spectrum but rather by the real-space spectrum of $h\left(L\right)$ versus $L$. We call this approach “hyperuniformity disorder length spectroscopy”.

Figure 1

Example binary pixel pattern, consisting of a $50\times 50$ grid of pixels and 125 particles (solid blue) occupying an area fraction of $\varphi =125/(50\times 50)=0.05$.

Figure 2

Example “Einstein pattern” of pixel particles with a Gaussian-distributed random displacement from a triangular lattice. The lattice constant is 30 pixels, and the root-mean-square displacement is 150 pixels in each dimension. Since the particles are effectively bound to the lattice sites, the pattern is hyperuniform with no long-range density fluctuations—even though it appears very disordered to the eye. Randomly placed measurement windows hence show density fluctuations that are due only to particles within a distance $h$ of the window boundary that is constant for large windows. Here this “hyperuniformity disorder length” is $h=82$ pixels, as illustrated for an example $1100\times 1100$ window. For clarity, the pixel particles are shown as dots that are 11 pixels across.

Figure 3

Area fraction variance (a) and hyperuniformity disorder length (b) versus measuring window size, for simulated two-dimensional random binomial pixel patterns with total area fraction $\varphi$ as labeled. The system size is $8600p_o\times 8600p_o$, where $p_o$ is the pixel width; this is indicated by yellow shading. The results agree well with expectation of (a) ${{\sigma }_{\varphi }}^2/\left[\varphi (1-\varphi )\right]={(p_o/L)}^2$ and (b) $h=L/2$. However, there are significant finite-size effects for large $L$, so all further simulated patterns will be analyzed only up to half the system size.

Figure 4

Area fraction variance (a), variance ratio (b), and hyperuniformity disorder length (c) versus measuring window size, for simulated two-dimensional triangular crystals with a specified fraction $f$ of random vacancies. Here $\varphi$ is the area fraction, $p_o$ is the pixel width, $8600p_o$ is image width (yellow shading), and $b=30p_o$ is the lattice spacing. Measurements are made at all integer $L/p_o$ for $f=0$ (gray curve), but only at a select subset for $f>0$. Results for $f=0$ at the same subset of $L/p_o$ are shown by black curves. The dash-dotted curves correspond to our model, Eq. (31); these curves appear to stop below a certain $L$, but in fact they are present but entirely covered by the data curve. The red and green dashed curves represent the upper and lower bounds given by Eqs. (28, 29, 30). The separated-particle bound is shown only for the crystal.

Figure 5

Example Einstein patterns for different root-mean-square displacements, labeled by the value of $\delta =x_{\text{rms}}/b=y_{\text{rms}}/b$. Here the lattice spacing $b$ is 30 pixels, and for clarity the particles are shown as dots that are many pixels across. By eye, patterns (f)–(i) are nearly indistinguishable and appear quite random.

Figure 6

Area fraction variance (a), variance ratio (b), and hyperuniformity disorder length (c) versus measuring window size, for simulated two-dimensional Einstein patterns with specified dimensionless root-mean-square displacement, $\delta$. Here $\varphi$ is the area fraction, $p_o$ is the pixel width, $8600p_o$ is image width (yellow shading), and $b=30p_o$ is the lattice spacing. In (c) an empirical fitting function is shown, which may be used to estimate the constant value $h_e$ of the hyperuniformity length at large $L$. Note that $h_e$ is roughly one-half the root-mean-square displacement in each dimension. Measurements are made at all integer $L/p_o$ for $\delta =0$ (gray curve), but only at a select subset for $\delta >0$. Results for $\delta =0$ at the same subset of $L/p_o$ are shown by black curves. The red and green dashed curves represent the upper and lower bounds given by Eqs. (28, 29, 30).

Figure 7

Large-$L$ value of the hyperuniformity disorder length versus dimensionless root-mean-square displacement, $\delta =x_{\text{rms}}/b=y_{\text{rms}}/b$, for the Einstein pattern results shown in Fig. 6 with lattice spacing $b=30p_o$. The insert is a blowup of the small-$\delta$ region. The fitting function $h_e=\sqrt{{(p_o/2)}^2+{\left(cb\delta \right)}^2}$ matches the data well with $c=0.56\pm 0.01$. By contrast a linear fit for all $\delta$, shown by the dashed yellow line in the inset, fails for small $\delta$.

Figure 8

Ratio of simulated to expected area fraction variance versus $L/W$ where $L$ is the width of the measuring boxes and $W=8300p_o$ is the width of the sample. Data are from previous figures. As seen, the effect of finite sample size is to suppressed the variance at large $L$. The longer the range of the density fluctuations, the larger the effect.

Figure 9

Normalized area fraction variance versus measuring window size for four random multinomial pixel patterns. Two mixtures of pixel particle species are used, as labeled, and patterns for each are created with total area fractions of 0.1 (smaller symbols) and 0.9 (larger symbols). A small portion of the $\{1,2,3\}$ system with $\varphi =0.9$ is shown in the inset. For all four patterns, the relative variance results agree well with the ${(p_o/L)}^2$ expectation shown by the dashed red line.

Figure 10

Relative variance versus measuring window size for a random arrangement of a 50:50 bidisperse mixture of extended square particles (gray squares) and the corresponding central-pixel representation (red dots). The actual image sizes are $4300p_o\times 4300p_o$; much smaller versions are shown as insets. Particle sizes and total area fractions are labeled. The ${\varphi }_i$-weighted average area of the two species is $\langle a\rangle =340{p_o}^2$. The red line is the $\langle a\rangle /L^2$ expectation. The gray curve is from Ref. [29].

Figure 11

Relative variance (a), variance ratio (b), and hyperuniformity disorder length (c) versus measuring window size for the central-pixel representation of arrangements of nonoverlapping $a_1={\left(10p_o\right)}^2$ and $a_2={\left(20p_o\right)}^2$ square particles, in equal numbers, like in the inset, for several area fractions. For clarity, error bars are plotted on only a few data as sets marked in the legend. The red and green broken curves represent the upper and lower bounds given by Eqs. (28, 29, 30). The separated-particle bound is shown only for the largest packing fraction.