# Structural Characterization of Many-Particle Systems on Approach to
Hyperuniform States

Salvatore Torquato  Email<EMAIL_ADDRESS>Department of Chemistry,
Department of Physics, Princeton Institute for the Science and Technology of
Materials, and Program in Applied and Computational Mathematics, Princeton
University, Princeton, New Jersey 08544, USA

###### Abstract

The study of hyperuniform states of matter is an emerging multidisciplinary
field, impinging on topics in the physical sciences, mathematics and biology.
The focus of this work is the exploration of quantitative descriptors that
herald when a many-particle system in $d$-dimensional Euclidean space
$\mathbb{R}^{d}$ approaches a hyperuniform state as a function of the relevant
control parameter. We establish quantitative criteria to ascertain the extent
of hyperuniform and nonhyperuniform distance-scaling regimes as well as the
crossover point between them in terms of the “volume” coefficient $A$ and
“surface-area” coefficient $B$ associated with the local number variance
$\sigma^{2}(R)$ for a spherical window of radius $R$. The larger the ratio
$B/A$, the larger the hyperuniform scaling regime, which becomes of infinite
extent in the limit $B/A\to\infty$. To complement the known direct-space
representation of the coefficient $B$ in terms of the total correlation
function $h({\bf r})$, we derive its corresponding Fourier representation in
terms of the structure factor $S({\bf k})$, which is especially useful when
scattering information is available experimentally or theoretically. We also
demonstrate that the free-volume theory of the pressure of equilibrium
packings of identical hard spheres that approach a strictly jammed state
either along the stable crystal or metastable disordered branch dictates that
such end states be exactly hyperuniform. Using the ratio $B/A$, as well as
other diagnostic measures of hyperuniformity, including the hyperuniformity
index $H$ and the direct-correlation function length scale $\xi_{c}$, we study
three different exactly solvable models as a function of the relevant control
parameter, either density or temperature, with end states that are perfectly
hyperuniform. Specifically, we analyze equilibrium systems of hard rods and
“sticky” hard-sphere systems in arbitrary space dimension $d$ as a function of
density. We also examine low-temperature excited states of many-particle
systems interacting with “stealthy” long-ranged pair interactions as the
temperature tends to zero, where the ground states are disordered,
hyperuniform and infinitely degenerate. We demonstrate that our various
diagnostic hyperuniformity measures are positively correlated with one
another. The same diagnostic measures can be used to detect the degree to
which imperfections in nearly hyperuniform systems cause deviations from
perfect hyperuniformity. Moreover, the capacity to identify hyperuniform
scaling regimes should be particularly useful in analyzing experimentally- or
computationally-generated samples that are necessarily of finite size.

##  I Introduction

A hyperuniform point configuration in $d$-dimensional Euclidean space
$\mathbb{R}^{d}$ is characterized by an anomalous suppression of large-scale
density fluctuations relative to those in typical disordered systems, such as
liquids and structural glasses Torquato and Stillinger (2003); Torquato
(2018). More precisely, a hyperuniform point pattern is one in which the
structure factor $S({\bf k})\equiv 1+\rho\tilde{h}(\bf{k})$ tends to zero as
the wavenumber $k\equiv|\bf{k}|$ tends to zero Torquato and Stillinger (2003);
Torquato (2018), i.e.,

$\displaystyle\lim_{|{\bf k}|\to 0}S({\bf k})=0,$ (1)

where $\tilde{h}(\bf{k})$ is the Fourier transform of the total correlation
function $h({\bf r})\equiv g_{2}(\bf{r})-1$ and $g_{2}({\bf r})$ is the pair-
correlation function Hansen and McDonald (2013). The hyperuniformity concept
generalizes the traditional notion of long-range order in many-particle
systems to include all perfect crystals, perfect quasicrystals, and exotic
amorphous states of matter. Disordered hyperuniform materials can have
advantages over crystalline ones, such as unique or nearly optimal, direction-
independent physical properties and robustness against defects Florescu  _et
al._ (2009); De Rosa  _et al._ (2015); Leseur  _et al._ (2016); Ma  _et al._
(2016); Zhang  _et al._ (2016a); Gkantzounis  _et al._ (2017); Froufe-Pérez
_et al._ (2017); Chen and Torquato (2018); Zhang  _et al._ (2018); Gorsky  _et
al._ (2019); Sheremet  _et al._ (2020); Kim and Torquato (2020); Yu  _et al._
(2021).

An equivalent definition of hyperuniformity is based on the local number
variance $\sigma^{2}(R)\equiv\langle N(R)^{2}\rangle-\langle N(R)\rangle^{2}$
associated with the number $N(R)$ of points within a $d$-dimensional spherical
observation window of radius $R$, where angular brackets denote an ensemble
average. A point pattern in ${\mathbb{R}}^{d}$ is hyperuniform if its variance
grows in the large-$R$ limit slower than $R^{d}$. This behavior is to be
contrasted with those of typical disordered systems, such as Poisson point
patterns, gases and liquids, where the number variance scales like the volume
$v_{1}(R)$ of the observation window, which is given by

$v_{1}(R)=\frac{\pi^{d/2}R^{d}}{\Gamma(1+d/2)}.$ (2)

Consider systems that are characterized by a structure factor with a radial
power-law form in the vicinity of the origin, i.e.,

$\displaystyle S({\bf k})\sim|{\bf k}|^{\alpha}\quad\text{for }|{\bf k}|\to
0.$ (3)

For hyperuniform systems, the exponent $\alpha$ is positive ($\alpha>0$) and
its value determines three different large-$R$ scaling behaviors of the number
variance Torquato and Stillinger (2003); Zachary and Torquato (2009); Torquato
(2018):

$\displaystyle\sigma^{2}(R)$ $\displaystyle\sim\left\\{\begin{array}[]{l
l}R^{d-1},&\alpha>1\text{ (class I)}\\\ R^{d-1}\ln R,&\alpha=1\text{ (class
II)\,.}\\\ R^{d-\alpha},&\alpha<1\text{ (class III)}\end{array}\right.$ (7)

These scalings of $\sigma^{2}(R)$ define three classes of hyperuniformity
Torquato (2018), with classes I and III describing the strongest and weakest
forms of hyperuniformity, respectively. States of matter that belong to class
I include all perfect crystals Torquato and Stillinger (2003); Zachary and
Torquato (2009), many perfect quasicrystals Zachary and Torquato (2009); Lin
_et al._ (2017); Oğuz  _et al._ (2017), and “randomly” perturbed crystal
structures Gabrielli (2004); Gabrielli and Torquato (2004); Gabrielli  _et
al._ (2008); Kim and Torquato (2018), classical disordered ground states of
matter Torquato and Stillinger (2003); Uche  _et al._ (2004); Torquato  _et
al._ (2015) as well as systems out of equilibrium Zhang  _et al._ (2016b); Lei
and Ni (2019). Class II hyperuniform systems include some quasicrystals Oğuz
_et al._ (2017), the positions of the prime numbers Torquato  _et al._ (2019),
and many disordered classical Donev  _et al._ (2005a); Zachary  _et al._
(2011); Jiao and Torquato (2011); Atkinson  _et al._ (2016); Zhang  _et al._
(2016b) and quantum Feynman and Cohen (1956); Reatto and Chester (1967);
Torquato  _et al._ (2008) states of matter. Examples of class III hyperuniform
systems include classical disordered ground states Zachary and Torquato
(2011), random organization models Hexner and Levine (2015) and perfect
glasses Zhang  _et al._ (2016b).

By contrast, for any nonhyperuniform system, it is shown in Appendix A that
the local variance has the following large-$R$ scaling behaviors:

$\displaystyle\sigma^{2}(R)\sim\begin{cases}R^{d},&\alpha=0^{+}\quad\text{(typical
nonhyperuniform)}\\\ R^{d-\alpha},&\alpha<0\quad\text{(anti-hyperuniform)}.\\\
\end{cases}$ (8)

For a “typical” nonhyperuniform system, $S(0)$ is bounded Torquato (2018). In
anti-hyperuniform systems, $S(0)$ is unbounded, i.e.,

$\lim_{|{\bf k}|\to 0}S({\bf k})=+\infty,$ (9)

and hence are diametrically opposite to hyperuniform systems. Anti-
hyperuniform systems include fractals, systems at thermal critical points
(e.g., liquid-vapor and magnetic critical points) Widom (1965); Kadanoff
(1966); Fisher (1967); Wilson and Kogut (1974); Binney  _et al._ (1992) as
well as certain substitution tilings Oğuz  _et al._ (2019).

Our main concern in this paper is the exploration of quantitative descriptors
that herald when a many-particle system is nearly hyperuniform or approaching
a hyperuniform state, whether ordered or not. Elucidating such questions not
only is expected to lead to a deeper fundamental understanding of the nature
and formation of hyperuniform systems but has great practical value. For
example, since hyperuniformity can endow a system with novel or optimal
physical properties, it is essential to know how close the system must be to
perfect hyperuniformity without significantly degrading its ideal performance.
In practice, perfect hyperuniformity is never achieved due to defects that are
inevitably present in any real finite-sized system Dreyfus  _et al._ (2015),
whether crystalline, quasicrystalline or disordered Kim and Torquato (2018).

We begin by recalling pertinent previous concepts and results (Sec. II),
including the fluctuation-compressibility theorem, hyperuniformity as a
critical phenomenon, a hyperuniformity length scale $\xi_{c}$ and a
hyperuniformity index $H$. The latter two quantities provide measures of
nearness to hyperuniformity.

In the remainder of the paper, we obtain a variety of theoretical results to
study the problem at hand. First, for a general system, we establish
quantitative criteria to ascertain the extent of hyperuniform and
nonhyperuniform distance-scaling regimes as well as the crossover point
between them in terms of the “volume” coefficient $A$ and “surface-area”
coefficient $B$ associated with the variance $\sigma^{2}(R)$ (Sec. III).
Specifically, the ratio $B/A$ determines the crossover length scale $R_{c}$.
The larger the ratio $B/A$, the larger the hyperuniform scaling regime, which
becomes of infinite extent in the limit $B/A\to\infty$. This capacity to
determine hyperuniform scaling regimes is expected to be particularly useful
in analyzing experimentally- or computationally-generated samples that are
necessarily of finite size.

Second, to complement the known direct-space representation of the coefficient
$B$ in terms of the total correlation function $h({\bf r})$ Torquato and
Stillinger (2003), we derive here its corresponding Fourier representation in
terms of the structure factor $S({\bf k})$ (Sec. IV). The latter
representation is particularly useful when the scattering intensity is
available experimentally or if the structure factor is known analytically.

Third, we show that the free-volume theory of the pressure of equilibrium
packings of identical hard spheres that approach either a strictly jammed
crystal or disordered state dictates that such jammed states be perfectly
hyperuniform (Sec. V). We describe why this outcome implies that such jammed
states must be defect-free.

Fourth, motivated by a desire to rely on analytical rather than numerical
methods, we structurally characterize three different disordered-system models
as a function of the relevant control parameter, either density or
temperature, with end states that are perfectly hyperuniform. These models are
distinguished from most other models with hyperuniform states in that their
pair correlation functions and structure factors are known exactly for all
values of the control parameter in the thermodynamic limit. We purposely avoid
the use of simulations of many-particle systems in finite boxes to draw
conclusions, since hyperuniformity in an infinite-wavelength property. The
first model that we characterize is an equilibrium system of hard rods (Sec.
VI). Here the control parameter is the number density $\rho$ (or packing
fraction) and its terminal value corresponds to the jammed state that is the
integer lattice. The second model studied is a certain “sticky” hard-sphere
system in arbitrary space dimension $d$ as a function of the number density
(or packing fraction) (Sec. VII). The third model that we characterize are
low-temperature excited states of so-called stealthy long-ranged pair
interactions in $\mathbb{R}^{d}$ (Sec. VIII). Here, the control parameter is
the temperature $T$ and the corresponding ground states at $T=0$ are
disordered and degenerate.

##  II Compressibility, Inverted Critical Point, Growing Length Scale and
Hyperuniformity Index

In this section, we briefly review pertinent background material. This outline
includes the implications of the fluctuation-compressibility theorem,
hyperuniformity as a critical phenomenon, a length scale that grows on
approach to a hyperuniform state, and the hyperuniformity index.

###  II.1 Fluctuation-Compressibility Theorem

Let us now recall the well-known fluctuation-compressibility theorem that
links the isothermal compressibility $\kappa_{T}$ of equilibrium single-
component many-particle ensembles at number density $\rho$ and temperature $T$
to infinite-wavelength density fluctuations Hansen and McDonald (2013). In
particular, for “open” systems in equilibrium, one has

$\rho k_{B}T\kappa_{T}=\frac{\langle N^{2}\rangle_{*}-\langle
N\rangle_{*}^{2}}{\langle N\rangle_{*}}=S({\bf
k=0})=1+\rho\int_{\mathbb{R}^{d}}h({\bf r})d{\bf r},$ (10)

where $k_{B}$ is Boltzmann’s constant and $\langle\rangle_{*}$ denotes an
average in the grand canonical ensemble.

The fluctuation-compressibility relation (10) enables one to draw useful
conclusions about the hyperuniformity of equilibrium systems. Any ground state
($T=0)$ in which the isothermal compressibility $\kappa_{T}$ is bounded and
positive must be hyperuniform because the structure factor $S({\bf k=0})$ must
be zero according to relation (10) Torquato (2018). More generally, we infer
from (10) that if the product $T\kappa_{T}$ tends to a nonnegative constant
$c$ in the limit $T\rightarrow 0$, then the ground-state of this system in
this zero-temperature limit must be nonhyperuniform if $c>0$ or hyperuniform
if $c=0$. By the same token, this means that increasing the temperature by an
arbitrarily small positive amount when a system is initially at a hyperuniform
ground state will destroy perfect hyperuniformity, since $S({\bf k=0})$ must
deviate from zero by some small amount determined by the temperature
dependence of the product $\kappa_{T}T$ for small $T$ Torquato (2018). This
indirectly implies that phonons or vibrational modes for sufficiently small
$T$ generally destroy the hyperuniformity of ground states Torquato  _et al._
(2015); Kim and Torquato (2018). Additionally, in order to have a hyperuniform
system that is in equilibrium at any positive $T$, the isothermal
compressibility must be zero, i.e., the system must be thermodynamically
incompressible Torquato (2018).

###  II.2 Inverted Critical Point and Scaling Laws

The direct correlation function $c({\bf r})$ of a hyperuniform system behaves
in an unconventional manner compared to that of typical liquids. This function
is defined via the Ornstein-Zernike integral equation Ornstein and Zernike
(1914):

$h({\bf r})=c({\bf r})+\rho c({\bf r})\otimes h({\bf r}),$ (11)

where $\otimes$ denotes a convolution integral. Fourier transforming (11) and
solving for ${\tilde{c}}({\bf k})$, the Fourier transform of $c(\bf r)$,
yields

${\tilde{c}}({\bf k})=\frac{{\tilde{h}}({\bf k})}{S({\bf
k})}=\frac{{\tilde{h}}({\bf k})}{1+\rho{\tilde{h}}({\bf k})}.$ (12)

By definition, a hyperuniform system is one in which ${\tilde{h}}({\bf
k=0})=-1/\rho$, i.e., the volume integral of $h(\bf r)$ exists, implying that
$h({\bf r})$ is sufficiently short-ranged in the sense that it decays to zero
faster than $|{\bf r}|^{-d}$. Interestingly, this means that the denominator
on the right side of (12) vanishes at ${\bf k=0}$ and therefore
${\tilde{c}}({\bf k=0})$ diverges to $-\infty$. This behavior implies that the
the volume integral of $c({\bf r})$ does not exist and hence the real-space
direct correlation function $c(\bf r)$ is long-ranged, i.e., decays slower
than $|{\bf r}|^{-d}$. We see that this behavior stands in diametric contrast
to standard thermal critical-point systems in which the total correlation
function is long-ranged and the direct correlation function is short-ranged
such that its volume integral exists Widom (1965); Kadanoff (1966); Fisher
(1967); Wilson and Kogut (1974). For this reason, it has been said that
hyperuniform systems are at an “inverted” critical point Torquato and
Stillinger (2003). As noted earlier, systems at thermal critical points are
anti-hyperuniform.

There is a class of disordered hyperuniform systems with concomitant critical
exponents Torquato and Stillinger (2003); Torquato (2018). For such
hyperuniform critical systems, the direct correlation function has the
following asymptotic behavior for large $r\equiv|{\bf r}|$ and sufficiently
large $d$:

$c({\bf r})\sim-\frac{1}{r^{d-2+\eta}}\qquad(r\rightarrow\infty),$ (13)

where $(2-d)<\eta\leq 2$ is a “critical” exponent associated with $c(\bf r)$
for hyperuniform systems that depends on the space dimension. The Fourier
transform of (13) yields

${\tilde{c}}({\bf k})\sim-\frac{1}{k^{2-\eta}}\qquad(k\rightarrow 0),$ (14)

which, when combined with (12), yields the asymptotic form of the structure
factor

$S({\bf k})\sim k^{2-\eta}\qquad(k\rightarrow 0),$ (15)

where $\eta=2-\alpha$ and $\alpha$ is the exponent defined in relation (3).

In what follows, it is assumed for concreteness, that the number density is
the control parameter. We define the following dimensionless density:

$\phi=\rho v_{1}(D/2),$ (16)

where $v_{1}(R)$ is given by (2) and $D$ is a characteristic “microscopic”
length scale. The direct correlation in the vicinity of a hyperuniform
critical state with dimensionless density $\phi_{c}$, i.e., for
$|\phi_{c}-\phi|\ll 1$, in sufficiently high dimensions has the following
large-$r$ asymptotic form Torquato and Stillinger (2003); Torquato (2018):

$c({\bf r})\sim\frac{\exp(-|{\bf r}|/\xi)}{|{\bf r}|^{d-2+\eta}},$ (17)

where $\xi$ is the correlation length. If the system approaches a hyperuniform
state from below the critical density $\phi_{c}$, the correlation length and
inverse of the structure factor at $k=0$, $S^{-1}(0)$, which is proportional
to ${\tilde{c}}(0)$, are described by the following scaling laws:

$\xi\sim\left(1-\frac{\phi}{\phi_{c}}\right)^{-\nu}\quad(\phi\rightarrow\phi_{c}^{-}),$
(18)
$S^{-1}(0)\sim\left(1-\frac{\phi}{\phi_{c}}\right)^{-\gamma},\quad(\phi\rightarrow\phi_{c}^{-}),$
(19)

where $\nu$ and $\gamma$ are nonnegative critical exponents. Observe that the
exponent $\gamma$ is a measure of how quickly a system approaches a critical
point. Combination of the three previous scaling laws leads to the following
interrelation between the exponents:

$\gamma=(2-\eta)\nu.$ (20)

The specific values of the critical exponents determine the universality class
of the hyperuniform system. It is noteworthy that all class III hyperuniform
systems are at critical points with the aforementioned scaling laws. Specific
examples of such a class III critical-point systems are nonequilibrium
absorbing-state models Hexner and Levine (2015).

###  II.3 Growing Length Scale

Another important length scale $\xi_{c}$ that can be extracted from the direct
correlation function and which grows as a hyperuniform state is approached is
defined by Hopkins  _et al._ (2012)

$\xi_{c}=[-{\tilde{c}}(k=0)]^{1/d}.$ (21)

Thus, we see that $\xi_{c}$ is the $d$th root of the volume integral of the
direct correlation function $c({\bf r})$. At a hyperuniform critical point,
$\xi_{c}$ diverges to $+\infty$. It was shown that a precursor to the
hyperuniform maximally random jammed (MRJ) state of sphere packings Torquato
_et al._ (2000) under compression was evident for densities far below the
jamming density was reached, as reflected by this static growing length scale
Hopkins  _et al._ (2012). The quantity $\xi_{c}$ was also used to identify
length scales in supercooled atomic liquid models that substantially grow as
the temperature decreased Marcotte  _et al._ (2013).

###  II.4 Hyperuniformity Index

The hyperuniformity index $H$ provides a measure of the nearness of a system
to a hyperuniform state, which is defined as

$H\equiv\frac{S(0)}{S(k_{p})},$ (22)

where $k_{p}$ is the wavenumber $k$ associated with the largest peak height of
the angularly averaged structure factor. One may empirically deem a system to
be nearly or effectively hyperuniform if $H$ is roughly less than about
$10^{-4}$ Atkinson  _et al._ (2016). The $H$ index has been profitably used to
quantify the effective hyperuniformity of polymer systems Chremos and Douglas
(2017, 2018), amorphous ices Martelli  _et al._ (2017), states along the
metastable extension of the hard-sphere systems away from jamming Zhou  _et
al._ (2020) and low-temperature states of “quantizer” systems Klatt  _et al._
(2019); Hain  _et al._ (2020).

##  III Hyperuniform and Nonhyperuniform Scaling Regimes

For a large class of ordered and disordered systems, the number variance
$\sigma^{2}(R)$ has the following large-$R$ asymptotic behavior Torquato and
Stillinger (2003); Torquato (2018):

$\sigma^{2}(R)=2^{d}\phi\left[A\left(\frac{R}{D}\right)^{d}+B\left(\frac{R}{D}\right)^{d-1}+o\left(\frac{R}{D}\right)^{d-1}\right]$
(23)

where $\phi$ is the dimensionless density given by (2) and
$o\left(R/D\right)^{d-1}$ represents terms of lower order than
$\left(R/D\right)^{d-1}$. Moreover, $A$ and $B$ are “volume” and “surface-
area” coefficients, respectively, which can be expressed as the following
volume integrals involving the total correlation function $h({\bf r})$,
respectively:

$\displaystyle A$ $\displaystyle=$ $\displaystyle\lim_{|\mathbf{k}|\rightarrow
0}S(\mathbf{k})=1+\rho\int_{\mathbb{R}^{d}}h({\bf r})d{\bf r}$ (24)
$\displaystyle=$ $\displaystyle
1+d\,2^{d}\,\phi\int_{0}^{\infty}x^{d-1}h(x)dx,$ $\displaystyle B$
$\displaystyle=$
$\displaystyle-\frac{d\,\Gamma(d/2)\rho}{2\,\pi^{1/2}\,D\Gamma[(d+1)/2]}\int_{\mathbb{R}^{d}}|{\bf
r}|h({\bf r})d{\bf r}$ (25) $\displaystyle=$
$\displaystyle-\frac{d^{2}\,2^{d-1}\,\Gamma(d/2)\phi}{\pi^{1/2}\Gamma[(d+1)/2]}\int_{0}^{\infty}x^{d}\,h(x)dx,$

and $x=r/D$ is a dimensionless distance. Here $h(r)$ is the radial function
that depends on the distance $r\equiv|{\bf r}|$, which results from averaging
the vector-dependent quantity $h({\bf r})$, i.e.,

$h(r)=\frac{1}{\Omega}\int_{\Omega}h({\bf r})\,d\Omega,$ (26)

$d\Omega$ is the differential solid angle and

$\Omega=\frac{d\pi^{d/2}}{\Gamma(1+d/2)}$ (27)

is the total solid angle contained in a $d$-dimensional sphere. In a perfectly
hyperuniform system Torquato and Stillinger (2003), the non-negative volume
coefficient vanishes, i.e., $A=0$, implying the sum rule

$\rho\int_{\mathbb{R}^{d}}h({\bf r})d{\bf r}=-1,$ (28)

such that the surface-area coefficient $B$ is nonnegative. Thus, such
hyperuniform systems fall within class I, since the variance grows like the
window surface area ($R^{d-1}$). On the other hand, when $A>0$ and $B=0$, the
system is hyposurficial, implying the sum rule

$\int_{0}^{\infty}x^{d}h(x)dx=0.$ (29)

Examples of hyposurficial systems include ideal gases, certain hard-core
systems in $\mathbb{R}^{d}$ Torquato and Stillinger (2003), non-equilibrium
phase transitions in amorphous ices Martelli  _et al._ (2017), and certain
systems with bounded pair interactions Zhang and Torquato (2020). Appendix A
provides a more general asymptotic expansion of the local number variance,
which is used to derive nonhyperuniform scaling laws.

For a large class of nonhyperuniform disordered systems that are sub-
Poissonian Torquato  _et al._ (2021), such as fluids and colloids without
particle clustering, the volume coefficient $A$ is often larger than the
magnitude of the surface-area coefficient $B$. For super-Poissonian
configurations Torquato  _et al._ (2021), such as the one described in
Appendix B, the magnitude of $B$ can be larger than $A$. When $A<1$ and $A<B$,
$B$ is often positive, and the smallness of the ratio $A/B$ measures the
degree of hyperuniformity Martelli  _et al._ (2017). In a disordered system
that is nearly hyperuniform, the inverse of this ratio, $B/A$ enables us to
ascertain hyperuniform and nonhyperuniform distance-scaling regimes of the
variance $\sigma^{2}(R)$ as a function of $R$. The crossover value of $R_{c}$
between these two scaling regimes is determined by equating the first two
terms of the large-$R$ asymptotic expansion of the local number variance, Eq.
(23), yielding the condition

$R_{c}\approx\frac{B}{A}.$ (30)

For the range $R_{0}<R<R_{c}$, where $R_{0}\approx\rho^{-1/d}$ (roughly, equal
to the mean-nearest neighbor distance), the system exhibits hyperuniform
scaling behavior, i.e., the variance is dominated by the surface-area scaling,
$R^{d-1}$. Clearly, the crossover value $R_{c}$ becomes infinite for perfectly
hyperuniform systems in which case $B/A=+\infty$. On the other hand, for
$R>R_{c}$ and finite $B/A$, the system begins to exhibit nonhyperuniform
scaling behavior, i.e., the variance is dominated by the surface-area scaling,
$R^{d}$. The determination of hyperuniform scaling regimes could be especially
useful in analyzing experimentally or computationally-generated systems that
are necessarily of finite size $L$ such that $R<L/4$ Torquato  _et al._
(2021).

##  IV Fourier Representation of Surface-Area Coefficient

Here we derive a Fourier representation of the surface-area coefficient $B$
for any homogeneous many-particle system, whether hyperuniform or not,
provided that the structure factor meets certain mild conditions. This
representation will be especially useful when the scattering intensity is
available experimentally or if the structure factor is analytically available,
as it is when solving the Ornstein-Zernike integral equation (11).

We define the Fourier transform of a function $f({\bf r})$ that depends on the
vector $\bf r$ in $d$-dimensional Euclidean space $\mathbb{R}^{d}$ as follows:

$\displaystyle\tilde{f}(\mathbf{k})=\int_{\mathbb{R}^{d}}f(\mathbf{r})\exp\left[-i(\mathbf{k}\cdot\mathbf{r})\right]d\mathbf{r},$
(31)

where $\bf k$ is a wave vector and
$(\mathbf{k}\cdot\mathbf{r})=\sum_{i=1}^{d}k_{i}r_{i}$ is the conventional
Euclidean inner product of two real-valued vectors. The function $f({\bf r})$
can generally represent a tensor of arbitrary rank. When it is well-defined,
the corresponding inverse Fourier transform is given by

$\displaystyle
f(\mathbf{r})=\left(\frac{1}{2\pi}\right)^{d}\int_{\mathbb{R}^{d}}\tilde{f}(\mathbf{k})\exp\left[i(\mathbf{k}\cdot\mathbf{r})\right]d\mathbf{k}.$
(32)

If $f$ is a radial function, i.e., $f$ depends only on the modulus
$r=|\mathbf{r}|$ of the vector $\bf r$, its Fourier transform is given by

$\displaystyle{\tilde{f}}(k)=\left(2\pi\right)^{\frac{d}{2}}\int_{0}^{\infty}r^{d-1}f(r)\frac{J_{\left(d/2\right)-1}\\!\left(kr\right)}{\left(kr\right)^{\left(d/2\right)-1}}\,dr,$
(33)

where $k=|{\bf k}|$ is the wavenumber or modulus of the wave vector $\bf k$
and $J_{\nu}(x)$ is the Bessel function of the first kind of order $\nu$. The
inverse transform of $\tilde{f}(k)$ is given by

$\displaystyle
f(r)=\frac{1}{\left(2\pi\right)^{\frac{d}{2}}}\int_{0}^{\infty}k^{d-1}\tilde{f}(k)\frac{J_{\left(d/2\right)-1}\\!\left(kr\right)}{\left(kr\right)^{\left(d/2\right)-1}}dk.$
(34)

The Fourier representation of the local number variance for statistically
homogeneous media, which includes perfect crystals (under uniform translations
of the crystals over their fundamental cells) Torquato (2018), is given by

$\sigma^{2}(R)=2^{d}\phi
R^{d}\Bigg{[}\frac{1}{(2\pi)^{d}}\int_{\mathbb{R}^{d}}S({\bf
k}){\tilde{\alpha}}({\bf k};{\bf R})d{\bf k}\Bigg{]},$ (35)

where

${\tilde{\alpha}}(k;R)=2^{d}\pi^{d/2}\Gamma(1+d/2)\frac{[J_{d/2}(kR)]^{2}}{k^{d}}$
(36)

is the Fourier transform of the scaled intersection volume function
$\alpha(r;R)$, which depends only on the magnitude of the wavevector $k=|\bf
k|$. Using the identity

$\frac{1}{(2\pi)^{d}}\int_{\mathbb{R}^{d}}{\tilde{\alpha}}(k;R)d{\bf k}=1,$
(37)

it follows from (35) that

$\sigma^{2}(R)=2^{d}\phi
S_{0}\,R^{d}+2^{d}\phi\,R^{d}\Bigg{[}\frac{1}{(2\pi)^{d}}\int_{\mathbb{R}^{d}}[S({\bf
k})-S_{0}]{\tilde{\alpha}}(k;R)d{\bf k}\Bigg{]},$ (38)

where

$S_{0}\equiv\lim_{|{\bf k}|\to 0}S({\bf
k})=A=1+\rho\int_{\mathbb{R}^{d}}h({\bf r})\,d{\bf r},$ (39)

which implies

$\rho{\tilde{h}}({\bf k=0})=S_{0}-1.$ (40)

Since ${\tilde{\alpha}}(k;R)$ is a radial function, depending only on the
magnitude of the wavevector, we can carry out the angular integration in the
integral in (38), yielding

$\displaystyle\sigma^{2}(R)$ $\displaystyle=$ $\displaystyle 2^{d}\phi
S_{0}\,R^{d}$ $\displaystyle+$ $\displaystyle
2^{d}\phi\,R^{d}\Bigg{[}\frac{d\pi^{d/2}}{(2\pi)^{d}\Gamma(1+d/2)}\int_{\mathbb{R}^{d}}k^{d-1}[S(k)-S_{0}]{\tilde{\alpha}}(k;R)dk\Bigg{]}.$
(41)

where the radial function $S(k)$ is given by

$S(k)=\frac{1}{\Omega}\int_{\Omega}S({\bf k})d\Omega.$ (42)

For large $R$,

${\tilde{\alpha}}(k;R)\sim
2^{d+1}\pi^{d/2-1}\Gamma(1+d/2)\frac{\cos^{2}[kR-(d+1)/4]}{Rk^{d+1}}.$ (43)

Combination of (41) and (43) yields the following large-$R$ asymptotic
expansion:

$\displaystyle\sigma^{2}(R)$ $\displaystyle\sim$ $\displaystyle 2^{d}\phi
S_{0}\,R^{d}$ $\displaystyle+$ $\displaystyle
2^{d}\phi\,R^{d-1}\Bigg{[}\frac{2d}{\pi}\int_{0}^{\infty}\frac{S(k)-S_{0}}{k^{2}}\cos^{2}[kR-\frac{d+1}{4}]\,dk\Bigg{]}\,$
(44) $\displaystyle+$ $\displaystyle{\cal O}(R^{d-3}).$

Using the identity

$\lim_{L\to\infty}\frac{1}{L}\int_{0}^{L}\cos^{2}[kR-\frac{d+1}{4}]\,dR=\frac{1}{2}$
(45)

and (44), we obtain

$\sigma^{2}(R)\sim 2^{d}\phi
S_{0}\,R^{d}+2^{d}\phi\,R^{d-1}\Bigg{[}\frac{d}{\pi}\int_{0}^{\infty}\frac{S(k)-S_{0}}{k^{2}}\,dk\Bigg{]}+{\cal
O}(R^{d-3}).$ (46)

Comparing (46) to (23) yields the desired Fourier representation of the
surface-area coefficient

$B=\frac{d}{\pi}\int_{0}^{\infty}\frac{S(k)-S_{0}}{k^{2}}\,dk.$ (47)

Thus, this Fourier representation of the coefficient $B$ is bounded provided
that the difference $[S(k)-S_{0}]$ tends to zero in the limit $k\to 0$ faster
than linear in $k$. Note that this condition will always be met by any
structure factor that is analytic at the origin, since $[S(k)-S_{0}]$ must
vanish at least as fast as quadratically in $k$ as $k\to 0$. Finally, we
observe that if the system is hyposurficial ($B=0$), relation (47) leads to
the integral condition

$\int_{0}^{\infty}\frac{S(k)-S_{0}}{k^{2}}\,dk=0,$ (48)

which is the Fourier-space sum rule for hyposurficality that corresponds to
the direct-space sum rule (29).

##  V Toward Jammed States in Equilibrium Hard-Sphere Systems

We recall some well-known results for the equilibrium phase behavior of
identical hard spheres of diameter $D$. The pressure of an equilibrium hard-
sphere system in any space dimension can be expressed in terms of the contact
values of the direct correlation function from the right and left sides via
the Ornstein–Zernike equation Torquato (2002):

$\frac{p}{\rho k_{B}T}=1+2^{d-1}\phi\,[c(D^{+})-c(D^{-})].$ (49)

Here the dimensionless density $\phi$ is to be interpreted as the packing
fraction, i.e., the fraction of space covered by the spheres. Figure 1
schematically shows the three-dimensional (3D) phase behavior in the
$\phi$-$p$ plane. Three different isothermal densification paths by which a
hard-sphere liquid may jam are shown. At sufficiently low densities, an
infinitesimally slow compression of the system at constant temperature defines
a thermodynamically stable liquid branch for packing fractions up to the
“freezing” point ($\phi\approx 0.494$). Increasing the density beyond the
freezing point putatively results in an entropy-driven first-order phase
transition Alder and Wainwright (1957); Frenkel (1999) to a crystal branch
that begins at the melting point ($\phi\approx 0.55$). Slow compression of the
system along the crystal branch ends at the jammed state corresponding to the
fcc lattice packing Mau and Huse (1999) with
$\phi=\pi/\sqrt{18}=0.74048\ldots$. Rapid compressions of the liquid while
suppressing some degree of local order can avoid crystal nucleation (on short
time scales) and produce a range of amorphous metastable extensions of the
liquid branch that jam only at the their density maxima. The faster the
compression rate, the lower is the jammed density. Presumably, the metastable
branch produced by the most rapid compression rate with a terminal density
consistent with strict jamming corresponds to the maximally random jammed
(MRJ) state with $\phi\approx 0.64$ Torquato  _et al._ (2000); Torquato and
Stillinger (2010). The MRJ state under the strict-jamming constraint Torquato
and Stillinger (2001); Donev  _et al._ (2004) is a prototypical glass Torquato
and Stillinger (2007) in that it is maximally disordered (according to a
variety of order metrics) without any long-range order (Bragg peaks) and
perfectly rigid, i.e., the elastic moduli are unbounded Torquato  _et al._
(2003); Torquato and Stillinger (2010).

Figure 1: The isothermal phase behavior of the 3D hard-sphere model in the
pressure-packing fraction plane, as adapted from Ref. Torquato and Stillinger,
2010. An infinitesimal compression rate of the liquid traces out the
thermodynamic equilibrium path (shown in red), including a discontinuity
resulting from the first-order freezing transition to a crystal branch (shown
in green) that ends at a maximally dense, infinite-pressure jammed state.
Rapid compressions of the liquid (blue curves) generate a range of amorphous
metastable extensions of the liquid branch that jam only at the their density
maxima, which we show here must be perfectly hyperuniform states.

In the immediate vicinity of a jammed state in $\mathbb{R}^{d}$ with packing
fraction $\phi_{J}$, the set of particle displacements that are accessible to
the packing approaches a convex-limiting polytope (because the impenetrability
constraints become linear) and free-volume theory Salsburg and Wood (1962);
Stillinger and Salsburg (1969); Donev  _et al._ (2005b) predicts that the
pressure $p$ has the following exact asymptotic form:

$\frac{p}{\rho k_{B}T}\sim\frac{d}{1-\phi/\phi_{J}}\qquad(\phi\to\phi_{J}).$
(50)

So far as the limiting polytope picture is concerned, the extremely narrow
connecting filaments that in principle connect the jamming neighborhoods have
so little measure that they do not overturn the free-volume leading behavior
of the pressure, even in the infinite-system limit Torquato and Stillinger
(2010). Although there is no rigorous proof yet for this claim, all numerical
evidence strongly suggests that it is correct as the jamming state is
approached either along the crystal Salsburg and Wood (1962); Stillinger and
Salsburg (1969) or metastable branch Donev  _et al._ (2005b). Assuming that
the system dynamics remain ergodic in the vicinity of a jammed state, we can
apply the fluctuation-compressibility relation (10) along with the free-volume
form (50) to yield the corresponding asymptotic relation for $S^{-1}(0)$:

$S^{-1}(0)\sim\frac{d}{(1-\phi/\phi_{J})^{2}}\qquad(\phi\to\phi_{J}).$ (51)

Thus, we see that in the limit $\phi\to\phi_{J}$, the system becomes perfectly
hyperuniform, which appears to validate a more general conjecture of Torquato
and Stillinger Torquato and Stillinger (2003), as elaborated below.

It is important to note that ergodicity condition required to obtain (51)
implies that the system must be defect-free. To stress this point further,
consider an fcc packing in $\mathbb{R}^{3}$ at the jammed state with packing
fraction $\phi=\pi/\sqrt{18}=0.74048\ldots$. Next, shrink each sphere by an
infinitesimal uniform amount (such that the impenetrability constraints are
linerizable) and randomly remove a small but statistically significant
fraction of spheres so that the resulting vacancies are well-separated from
one another. This vacancy-riddled packing remains in its jamming basin. Now,
slowly compress it until it jams, whereby the system pressure diverges. It is
known that the structure factor $S({\bf k})$ at the origin of a nearly
hyperuniform system with vacancies is proportional to the concentration of
vacancies Kim and Torquato (2018) and hence cannot be hyperuniform, as
predicted by (51). Thus, the free-volume form for the pressure (50) cannot
apply to this nonequilibrium vacancy-riddled but strictly jammed state.

More generally, Torquato and Stillinger Torquato and Stillinger (2003)
suggested that certain defect-free strictly jammed packings of identical
spheres are hyperuniform. Specifically, they conjectured that any strictly
jammed saturated infinite packing of identical spheres is hyperuniform. A
saturated packing of hard spheres is one in which there is no space available
to add another sphere. What is the rationale for such a conjecture? First, it
recognizes that mechanical rigidity is a necessary but not sufficient
condition for hyperuniformity. Indeed, requiring the saturation property in
the conjecture eliminates the class of strictly jammed crystal states that are
riddled with vacancies, as per the aforementioned example. Moreover, the
conjecture excludes packings that may have a rigid backbone but possess
“rattlers” (particles that are not locally jammed but are free to move about a
confining cage) because a strictly jammed packing, by definition, cannot
contain rattlers Donev  _et al._ (2005b); Torquato and Stillinger (2010).
Typical packing protocols that have generated disordered jammed packings tend
to contain a small concentration of rattlers Torquato and Stillinger (2010);
Parisi and Zamponi (2010), and hence the entire (saturated) packing cannot be
deemed to be jammed. Therefore, the conjecture cannot apply to current
numerically-generated disordered packings, even if the structure factor at the
origin is very small, e.g., an $H$ index of the order of $10^{-4}$ for 3D MRJ-
like sphere packing Donev  _et al._ (2005a). Thus, rattler-free disordered
jammed sphere packings are expected to be perfectly hyperuniform. Indeed, if
the free-volume theory applies along the metastable extension ending at the
MRJ state, rattlers cannot be present (due to a type of constrained ergodicity
on time scales much less than relaxation times) and hence should be perfectly
hyperuniform. It has been suggested that the ideal MRJ state is rattler-free,
implying that the packing is more disordered without the presence of rattlers
Atkinson  _et al._ (2016).

The consequences of the free-volume theory requires a modification of the
Torquato-Stillinger conjecture because the former eliminates defects of any
type in the jamming limit and the saturation condition may not prohibit all
defect types. For example, the saturation property may not exclude
dislocations in strictly jammed crystal states. A more refined variant of the
conjecture is the following statement: Any strictly jammed infinite packing of
identical spheres that is defect-free is hyperuniform.

In Appendix C, we obtain exact sum rules and the exact large-$k$ asymptotic
behaviors of the structure factors of certain general packings of identical
spheres in $\mathbb{R}^{d}$, whether in equilibrium or not.

##  VI Equilibrium Hard Rods

We first structurally characterize one-dimensional systems of identical hard
rods of length $D$ in equilibrium as a function of the packing fraction $\phi$
up to the jammed, hyperuniform state ($\phi=1$). This hyperuniform state is
the integer lattice $\mathbb{Z}$ and hence cannot be regarded to be a critical
hyperuniform state, as described in Sec. II.2.

Percus Percus (1964) obtained the following exact expression for the direct
correlation function for an equilibrium hard-rod system at packing fraction
$\phi$:

$c(r)=-\Theta(D-r)\frac{1-\phi r/D}{(1-\phi)^{2}}$ (52)

where $\Theta(x)$ is the Heaviside step function. Combination of this relation
with (49) yields the well-known expression for the pressure of the hard-rod
system Tonks (1936):

$\frac{p}{\rho k_{B}T}=\frac{1}{1-\phi},$ (53)

which we see takes the free-volume form (50) for the entire range of packing
fractions, not just near the jammed state $\phi_{J}=1$, which must be
hyperuniform. The Fourier transform of $c(r)$ is given by

${\tilde{c}}(k)=\frac{2\left[\phi\,(\cos(kD)-1)+kD\,\sin(kD)(\phi-1)\right]}{(1-\phi)^{2}(kD)^{2}}.$
(54)

Using (12), the corresponding structure factor is given by

$S(k)=\frac{1}{1-\frac{\displaystyle
2\,\phi\left[\phi\,(\cos(kD)-1)+kD\,\sin(kD)(\phi-1)\right]}{\displaystyle(1-\phi)^{2}(kD)^{2}}}.$
(55)

It is seen that the asymptotic large-$k$ behavior of $S(k)$ is consistent with
the general expression (107) for $d=1$, where the contact value
$g_{2}(D^{+})=(1-\phi)^{-1}$ is bounded for all $\phi$, except at its maximal
value $\phi=1$.

Figure 2: The structure factor $S(k)$ versus the dimensionless wavenumber $kD$
for a equilibrium hard-rod system near the jammed state ($\phi=0.98$). The
envelope of the largest peak values decays with wavenumber approximately as
the power law $1/k^{3/2}$.

The structure factor $S(k)$ is plotted in Fig. 2 for the hard-rod system near
the jammed state with $\phi=0.98$. At such high densities, its peak values
occur at wavenumbers that are approximately integer multiples of $2\pi/D$,
which is exactly the case for the integer lattice (jammed state) with lattice
constant $D$. The envelope of the largest peak values decays with wavenumber
approximately as the power law $1/k^{3/2}$. It immediately follows from (55)
that the volume coefficient $A$, defined by (24), is exactly given by

$A=S(0)=(1-\phi)^{2}$ (56)

The surface-area coefficient $B$ is easily obtained for all $\phi$ from its
Fourier representation (47) and use of the exact structure factor formula (55)
by high-precision numerical integration. These results for $B$ are in
excellent agreement with previous results using the direct-space
representation of $B$ Torquato and Stillinger (2003). A simple and highly
accurate formula for $B$ is given by

$B=\frac{\phi}{12}(6-8\phi+3\phi^{2}).$ (57)

This formula is obtained by using exact results at low and high packing
fractions and assuming that $B$ is a cubic polynomial in $\phi$ (without a
constant term). Specifically, the linear and quadratic terms are determined by
the exact expansion of $S(k)-S_{0}$ in powers of $\phi$ through second order
in $\phi$, as determined from the the exact formula (55). The remaining cubic
term is found by using the fact that $B$ is exactly equal to $1/12$ for the
integer lattice at $\phi=1$ Torquato and Stillinger (2003). Thus, combination
of the aforementioned formulas for $A$ and $B$, yields the following excellent
approximation for the ratio

$\frac{B}{A}=\frac{\phi(6-8\phi+3\phi^{2})}{12(1-\phi)^{2}}.$ (58)

The ratio $B/A$ is plotted in Fig. 3 as a function of $\phi$. It is seen that
$B$ is about ten times larger than $A$ at $\phi=0.9$ and the ratio
dramatically diverges to infinity as $\phi$ tends to unity. For example, at
$\phi=0.99$, $B$ is about 842 times larger than $A$. Referring to Sec. A, the
magnitude of the ratio $B/A$ determines the hyperuniformity scaling regime for
the local number variance $\sigma^{2}(R)$ to be those window radii in the
range ${\cal O}(\rho^{-1/d})<R<R_{c}=B/A$. For $R$ beyond ${\cal O}(B/A)$, the
variance should display nonhyperuniform scaling of the number variance. These
scaling regimes are verified in the plot of the variance at $\phi=0.99$ shown
in Fig. 4. The actual variance, as determined from the exact relations (35)
and (55), at such a high packing fraction is characterized by small-scale
fluctuations around some average values, and because such variations visually
obscure the scaling regimes, we considered employing the cumulative running
average, defined in Ref. Kim and Torquato (2017), to smooth out the small-
scale oscillations around the global values. Because the cumulative average
plot on the scales shown in Fig. 4 is very similar to one obtained from the
asymptotic expansion (23), we employ the later, for simplicity, to show the
scaling regimes.

Figure 3: The ratio of the surface-area coefficient to volume coefficient,
$B/A$, as a function of the packing fraction $\phi$ for equilibrium hard-rod
systems. Figure 4: The scaled variance $\sigma^{2}(R)/R^{2}$ versus $R/D$, as
computed from relation (23), for the jammed hyperuniform state with $\phi=1$
(as found in Ref. Torquato and Stillinger (2003)) and for a nonhyperuniform
state near jamming with $\phi=0.99$, where $R_{c}=B/A=841.7474996$. Because
the variance is bounded for the hyperuniform state Torquato and Stillinger
(2003), $\sigma^{2}(R)/R^{2}$ decreases with $R$ for large $R$ like $1/R^{2}$.

It easily follows from formula (55) that the largest peak value of the
structure factor $S(k_{p})$ through first-order in the packing fraction is
exactly given by

$\displaystyle S(k_{p})$ $\displaystyle=$ $\displaystyle
1-\frac{\sin(k_{p}D)}{k_{p}D}\phi+{\cal O}(\phi^{2})$ (59) $\displaystyle=$
$\displaystyle 1+(0.434464192\ldots)\phi+{\cal O}(\phi^{2}),$

where

$k_{p}D=\frac{3}{4}\pi+\frac{1}{4}\sqrt{9\pi^{2}-16}=4.489654702\ldots\,.$
(60)

It is highly nontrivial to derive an analytical expression for $S(k_{p})$ for
arbitrary $\phi$. Of course, we know that $S(k_{p})$ must increase as $\phi$
increases and diverges in the limit that the jamming density is approached
from below, i.e., $\phi\to 1^{-}$, which should be distinguished from the case
when $\phi=1$ in which $S(k)$ consists of Dirac delta functions located at
wavenumbers that are multiples of $2\pi/D$. The first peak value of the
structure factor, which is also the largest value, at a fixed value of the
packing fraction is plotted in Fig. 5 as a function of $\phi$, which of course
must diverge at the hyperuniform jammed state $\phi=1$. The following rational
function provides an excellent fit to the data for all $\phi$ up to the
jamming density:

$S(k_{p})=\frac{1+a_{1}\phi+a_{2}\phi^{2}+a_{3}\phi^{3}}{(1-\phi)^{2}},$ (61)

where $a_{1}=-1.565536$, $a_{2}=0.470399$ and $a_{3}=0.196196$. This fit uses
the exact small-$\phi$ expansion (59) to ascertain the coefficient $a_{1}$.
This accurate approximation when combined with exact formula (56) for $S(0)$
yields the following expression for the hyperuniformity index:

$H=\frac{(1-\phi)^{4}}{1+a_{1}\phi+a_{2}\phi^{2}+a_{3}\phi^{3}},$ (62)

which is plotted in Fig. 6 as function of $\phi$.

Figure 5: The largest peak value of the structure factor $S(k_{p})$ versus the
packing fraction $\phi$ for equilibrium hard-rod systems.  Figure 6: The
hyperuniform index $H$ versus the packing fraction $\phi$ for equilibrium
hard-rod systems. The rapid growth of $\xi_{c}$ with increasing packing
fraction is readily apparent.

Using (54), it immediately follows that the length scale $\xi_{c}$, defined by
(21), for equilibrium hard rods is given by

$\xi_{c}=-{\tilde{c}}(k=0)=\frac{(2-\phi)D}{(1-\phi)^{2}}.$ (63)

Figure 7 clearly shows that this length scale grows appreciably with
increasing packing fraction. It is already about an order of magnitude greater
than the length of a rod $D$ at $\phi=0.6$ and rapidly takes on even larger
values as $\phi$ increases further, eventually diverging to infinity in the
limit $\phi\to 1$.

Figure 7: The dimensionless length scale $\xi/D$ versus the packing fraction
$\phi$ for equilibrium hard-rod systems.

It is useful to compare and contrast the growth rate of the three descriptors
$B/A$, $\xi_{c}$ and $H^{-1}$ as the packing fraction $\phi$ increases and
approaches the jammed hyperuniform state. We see that the numerators of the
formulas (58) and (63) for moderate to high values of $\phi$ are of order
unity, and hence $B/A$ and $\xi_{c}$ have the same scaling behavior as the
hyperuniform state of the hard-rod system is approached, namely,

$\frac{B}{A}\sim\frac{\xi_{c}}{D}\sim\frac{1}{(1-\phi)^{2}}.$ (64)

This is to be contrasted with the inverse of the hyperuniformity index
$H^{-1}$, which grows exponentially faster, i.e., we see from (6) that it
grows like the square of the other two quantities:

$H^{-1}\sim\left(\frac{B}{A}\right)^{2}\sim\left(\frac{\xi_{c}}{D}\right)^{2}\sim\frac{1}{(1-\phi)^{4}}.$
(65)

##  VII Sticky Hard Spheres

We utilize a particular $g_{2}$-invariant process to analytically study the
approach of another disordered system to a hyperuniform but unjammed state as
a function of the relevant control parameter, the number density. A
$g_{2}$-invariant process is one in which a chosen nonnegative form for the
pair correlation function $g_{2}$ remains invariant over a nonvanishing
density range while keeping all other relevant macroscopic variables fixed
Torquato and Stillinger (2002, 2003). The upper limiting “terminal” density is
the point above which the nonnegativity condition on the structure factor,
i.e., $S({\bf k})\geq 0$ for all $\bf k$, would be violated. Thus, at the
terminal or critical density, the system is hyperuniform if the minimum in
$S({\bf k})$ occurs at the origin and it is realizable. This optimization
problem has deep connections to the sphere-packing problem. Specifically, it
is the linear program (LP) lower bound on the maximal packing density that is
dual to the Elkies-Cohn LP upper bound formulation Cohn and Elkies (2003). A
certain $g_{2}$-invariant test function was employed to conjecture that the
densest sphere packings in very high space dimensions are disordered, rather
than ordered as in low dimensions Torquato and Stillinger (2006).

Here we specifically consider the $g_{2}$-invariant process defined by a pair
correlation function that imposes a hard-core constraint via a unit step
function plus a delta function contribution that acts at contact $r=D$:

$g_{2}(r)=\Theta(r-D)+\frac{Z}{\rho s_{1}(D)}\delta(r-D),$ (66)

where $Z$ is the nonnegative average contact coordination number and

$s_{1}(r)=\frac{d\pi^{d/2}r^{d-1}}{\Gamma(1+d/2)}$ (67)

is the surface area of a sphere of radius $r$. It was previously shown that
this “sticky-sphere” pair correlation function is numerically realizable, to
an excellent approximation, up to the hyperuniform terminal packing fraction
for $d=2$ (see Fig. 8), implying that such pair correlation functions are also
realizable in all higher dimensions, as dictated by the decorrelation
principle Torquato and Stillinger (2006).

Figure 8: A numerically generated two-dimensional configuration of 500
“sticky” hard disks that realizes the step plus delta function pair
correlation function at the terminal packing fraction $\phi_{c}=1/2$, as
adapted from Ref. Uche  _et al._ (2006). This critical hyperuniform state
consists only of dimers, i.e., $Z=1$, and hence is unjammed, in contrast to
hyperuniform states in the phase diagram of hard spheres, as shown in Fig. 1.

Here we collect previously established key results for this $g_{2}$-invariant
process Torquato and Stillinger (2002, 2003) in order to characterize how such
systems for any $d$ approach a hyperuniform state as the packing fraction
increases up to the critical terminal value $\phi_{c}$. It was shown that

$Z=\frac{2^{d}d}{d+2}\phi$ (68)

is obeyed in order to constrain the location of the minimum of the structure
factor to be at ${\bf k=0}$, where the packing $\phi$ lies in the range
$0\leq\phi\leq\phi_{c}$, and

$\phi_{c}=\frac{d+2}{2^{d+1}}$ (69)

is the terminal or critical packing fraction, which has the same asymptotic
form as a lower bound on the maximal packing density for lattice packings
derived by Ball Ball (1992). At the hyperuniform critical point, the contact
number $Z_{c}=d/2$, indicating that such sticky-sphere systems are never
jammed in any dimension.

The corresponding structure factor for any $d$ and $\phi$ in the range
$0\leq\phi\leq\phi_{c}$ is given by

$S(k)=1+\frac{2^{d/2}\Gamma(2+d/2)}{(kD)^{(d/2)-1}}\left(\frac{\phi}{\phi_{c}}\right)\Bigg{[}\frac{J_{(d/2)-1}(kD)}{d+2}-\frac{J_{d/2}(kD)}{kD}\Bigg{]}.$
(70)

An important mathematical property of the structure factor for such sticky
hard spheres is that its extrema for a fixed $d$ are independent of the
packing fraction. For example, for $d=2$ and $d=5$, $k_{p}D=6.38015\ldots$ and
$k_{p}D=8.18255\ldots$, respectively. It is seen that the asymptotic large-$k$
behavior of $S(k)$ is consistent with the general expression (111) for any
$d$, where $Z$ is given by (68). Relation (70) leads to the power law

$S^{-1}(0)=\left(1-\frac{\phi}{\phi_{c}}\right)^{-1},\qquad\phi\rightarrow\phi_{c}^{-},$
(71)

which upon comparison to (19) yields the critical exponent $\gamma=1$. Figure
9 shows the structure factor $S(k)$ versus $kD$ for sticky hard spheres for
$d=2$ and $d=5$ at their respective hyperuniform states. We see that structure
factor reflects strong decorrelation in going from two to five dimensions,
which is consistent with the decorrelation principle Torquato and Stillinger
(2006).

Figure 9: The structure factor $S(k)$ versus the dimensionless wavenumber $kD$
for sticky hard spheres in two and five dimensions at their respective
hyperuniform states, i.e., $\phi=\phi_{c}$, where $\phi_{c}$ is given by (69).

In any space dimension $d$, the volume and surface-area coefficients for
$0\leq\phi\leq\phi_{c}$ are respectively given by

$A=S(k=0)=1-\frac{\phi}{\phi_{c}}$ (72)

and

$B=\frac{d^{2}\Gamma(d/2)}{8\,\sqrt{\pi}\,\Gamma((d+3)/2)}\frac{\phi}{\phi_{c}}.$
(73)

Thus, ratio $B/A$ is given by

$\frac{B}{A}=\frac{d^{2}\Gamma(d/2)}{8\,\sqrt{\pi}\,\Gamma((d+3)/2)}\frac{\phi/\phi_{c}}{(1-\phi/\phi_{c})}.$
(74)

We see that apart from $d$-dimensional constants, the behavior of the ratio
$B/A$ is exactly the same across dimensions when the packing fraction $\phi$
is scaled by the terminal packing fraction $\phi_{c}$. The ratio $B/A$ versus
$\phi$ is plotted in Fig. 10 for $d=2$ and $d=5$ for all $\phi$ up to their
respective critical points.

Figure 10: The ratio of the surface-area coefficient to volume coefficient,
$B/A$, as a function of the packing fraction $\phi$ for sticky hard spheres in
two and five dimensions.

We see from Fig. 9 that the peak value of the structure factor is not strongly
dependent on the space dimension and is of order unity. Thus, the
hyperuniformity index $H$ is primarily determined by the inverse of $A$ [cf.
72)] and so

$H\sim\frac{1}{1-\phi/\phi_{c}}.$ (75)

The actual values of $H$ are plotted in Fig. 11 as a function of $\phi$ for
$d=2$ and $d=5$ up to their respective critical points.

Figure 11: The hyperuniformity index $H$ as a function of the packing fraction
$\phi$ for sticky hard spheres in two and five dimensions.

Combination of (12) and (70) yields the following expression for the the
Fourier transform of the direct correlation for sticky spheres:

${\tilde{c}}(k)=\frac{\displaystyle\frac{(2\pi)^{d/2}D^{d}}{(kD)^{(d/2)-1}}\Bigg{[}\frac{\displaystyle
J_{(d/2)-1}(kD)}{d+2}-\frac{\displaystyle
J_{d/2}(kD)}{kD}\Bigg{]}}{\displaystyle
1+\frac{2^{d/2}\Gamma(2+d/2)}{(kD)^{(d/2)-1}}\left(\frac{\phi}{\phi_{c}}\right)\Bigg{[}\frac{J_{(d/2)-1}(kD)}{d+2}-\frac{J_{d/2}(kD)}{kD}\Bigg{]}}$
(76)

It immediately follows that at zero wavenumber, we have

${\tilde{c}}(0)=-2v_{1}(D)\left(1-\frac{\phi}{\phi_{c}}\right)^{-1},$ (77)

and, hence, the length scale defined by (21) yields

$\xi_{c}=[2v_{1}(D)]^{1/d}\left(1-\frac{\phi}{\phi_{c}}\right)^{-1/d},$ (78)

indicating that it grows more slowly as the space dimension increases. Figure
7 shows how the length scale $\xi_{c}$ grows with $\phi$ for $d=2$ and $d=3$.

The length scale $\xi_{c}$ should be contrasted with the correlation length
$\xi$, defined by (18), which, when combined with relation (77), yields

$\displaystyle\xi$ $\displaystyle=$
$\displaystyle\frac{D}{[8(d+2)(d+4)\phi_{c}]^{1/4}}\left(1-\frac{\phi}{\phi_{c}}\right)^{-1/4},\qquad\phi\rightarrow\phi_{c}^{-}.$
(79)

Comparison of (79) to the power law (18) yields the exponent $\nu=1/4$. It
should not go unnoticed that since the exponents $\gamma$, $\eta$ and $\nu$
are either integers or a simple fraction, independent of dimension, these
systems always behave in a mean-field manner at their critical points.

Figure 12: The dimensionless length scale $\xi_{c}/D$ versus the packing
fraction $\phi$ for sticky hard spheres in two and five dimensions.

All three descriptors $B/A$, $H^{-1}$ and $\xi_{c}$ are increasing functions
of the packing fraction $\phi$, but how do they correlate with one another?
From formulas (74) and (75), we see $B/A$ and $H^{-1}$ have the same scaling
behavior as the critical hyperuniform state of the sticky hard-system is
approached, namely,

$\frac{B}{A}\sim H^{-1}\sim\frac{1}{(1-\phi/\phi_{c})}.$ (80)

According to (78), the length scale $\xi_{c}$ for $d\geq 2$ grows more slowly
than the other two quantities as the critical state is approached, i.e.,

$(\xi_{c})^{d}\sim\frac{B}{A}\sim H^{-1}.$ (81)

##  VIII Stealthy Disordered Ground States

We are also interested in theoretically understanding how vibrational modes in
excited low-temperatures states of matter impact the approach to ground
states, which are necessarily hyperuniform (see Sec. II.1), as the temperature
$T$ tends to zero. For this purpose, we consider stealthy hyperuniform many-
particle systems Uche  _et al._ (2004); Batten  _et al._ (2008), which are a
subclass of class I hyperuniform systems Torquato  _et al._ (2015) in which
the structure factor is zero for a range of wavevectors around the origin,
i.e.,

$S({\bf k})=0\qquad 0\leq|{\bf k}|\leq K,$ (82)

where $K$ is some positive constant. Such systems are called “stealthy”
because they completely suppress single scattering of incident radiation for
the wave vectors within an exclusion sphere of radius $K$ centered at the
origin in Fourier space. It has been shown that certain bounded (soft) long-
ranged pair interactions have classical ground states that are stealthy and
hyperuniform Torquato  _et al._ (2015). The nature of the ground-state
configuration manifold (e.g., the degree of order) associated with such
stealthy pair potentials depends on the number of constrained wave vectors.
The dimensionless parameter $\chi$ measures the relative fraction of
constrained degrees of freedom compared to the total number of degrees of
freedom and, counterintuitively, is inversely proportional to the number
density Torquato  _et al._ (2015); specifically, it is explicitly given by the
formula

$\rho\,\chi=\frac{v_{1}(K)}{2d\,(2\pi)^{d}},$ (83)

where $v_{1}(K)$ the volume of a $d$-dimensional sphere of radius $K$ [cf. Eq.
(2)]. For $d=2$ and $d=3$, the ground states are disordered for
$0\leq\chi<1/2$. At $\chi=1/2$, there is a phase transition to a crystal
phase. The “collective-coordinate” optimization procedure represents a
powerful approach to generate numerically disordered stealthy hyperuniform
many-particle systems Uche  _et al._ (2004); Batten  _et al._ (2008); Torquato
_et al._ (2015).

A statistical-mechanical theory for disordered stealthy ground states has been
formulated in the canonical ensemble in the zero-temperature limit. By
exploiting an ansatz that the entropically favored (most probable) stealthy
ground states in the canonical ensemble behave like “pseudo” equilibrium hard-
sphere systems in Fourier space with an “effective packing fraction” $\phi$
that is proportional to $\chi$, one can employ well-established integral-
equation formulations for the pair correlation function of hard spheres in
direct space Hansen and McDonald (2013); Torquato (2002) to obtain accurate
theoretical predictions for $g_{2}({\bf r};\phi)$ and $S({\bf k};\chi)$ for a
moderate range of $\chi$ about $\chi=0$ Torquato  _et al._ (2015), i.e.,

$S({\bf k};\chi)=g_{2}^{HS}({\bf r=k};\phi).$ (84)

For example, the total correlation function $h(r)$ in the limit
$\chi\rightarrow 0$ for any $d$, which is exactly given by

$\rho h(r)=-\left(\frac{K}{2\pi
r}\right)^{d/2}J_{d/2}(Kr)\qquad(\chi\rightarrow 0),$ (85)

corresponds to the following unit-step function for the structure factor:

$S({\bf k})=\Theta(k-K)\qquad(\chi\rightarrow 0).$ (86)

Our interest in this paper are the excited states associated with the long-
ranged stealthy potentials Torquato  _et al._ (2015) close to the ground
states, i.e., when the absolute temperature $T$ is small. It was shown that
the structure factor at the origin $S(0)$ varies linearly with $T$ for excited
states. Importantly, this positive value of $S(0)$ is the uniform value of
$S({\bf k})$ for $0\leq|{\bf k}|\leq K$ for the special case of the step-
function power-law potential for small $\chi$, which was verified by
simulation results in various dimensions Torquato  _et al._ (2015). Because we
are interested in qualitative trends as the disordered ground state is
approached as the temperature is decreased, it suffices for our purposes to
consider the following simple form for the structure factor at positive but
small temperatures and sufficiently small $\chi$:

$S({\bf k})=T^{*}\Theta(K-k)+\Theta(k-K),$ (87)

where $T^{*}$ is a dimensionless temperature that is much smaller than unity.
The form (87) is obtained by combining the aforementioned uniform value for
$0\leq|{\bf k}|\leq K$ with the ground state structure factor as $\chi$ tends
to zero, i.e., Eq. (86).

It follows from (24) and (87) that the volume coefficient is given by

$A=T^{*}$ (88)

Moreover, substitution of (87) into the Fourier representation of the surface-
area coefficient (47) yields

$B=\frac{d}{\pi K}(1-T^{*})$ (89)

Thus, ratio $B/A$ is given by

$\frac{B}{A}=\frac{d}{\pi K}\frac{(1-T^{*})}{T^{*}},$ (90)

which is plotted in Fig. 13 as a function of the inverse of $T^{*}$ for
selected space dimensions. We see that at fixed temperature, the ratio $B/A$
increases with the space dimension. From relation (87), we see that the peak
value of the structure factor is trivially unity ($S(k_{p})=1$), and hence the
hyperuniformity index is simply linear in $T^{*}$, i.e.,

$H=T^{*}.$ (91)

Figure 13: The ratio of the surface-area coefficient to volume coefficient,
$B/A$, as a function of the inverse of the dimensionless temperature $1/T^{*}$
for excited states associated with the long-ranged stealthy pair potential for
dimensions $d=1,3$ and 6.

Combination of (12) and (87) yields the following expression for the the
Fourier transform of the direct correlation function for the excited states:

$\rho{\tilde{c}}({\bf k})=(1-T^{*})\Theta(K-k).$ (92)

Thus, evaluating this relation at zero wavenumber and use of (12) yields the
length scale

$\xi_{c}=(2\pi)\left[\frac{2d\,\chi}{v_{1}(K)}\right]^{1/d}\left(\frac{1}{T^{*}}-1\right)^{1/d}.$
(93)

This length scale is plotted in Fig. 14 as a function of the inverse
temperature for selected space dimensions.

Figure 14: The dimensionless length scale $\xi_{c}K$ versus the inverse of the
dimensionless temperature $1/T^{*}$ for excited states associated with the
long-ranged stealthy pair potential for dimensions $d=1,3$ and 6.

The three descriptors $B/A$, $H^{-1}$ and $\xi_{c}$ are increasing functions
of the inverse temperature. From formulas (90) and (75), we observe that $B/A$
and $H^{-1}$ have the same scaling behavior as the inverse temperature tends
to infinity, i.e., the disordered hyperuniform ground state is approached,
namely,

$\frac{B}{A}\sim H^{-1}\sim\frac{1}{T^{*}}.$ (94)

This is to be contrasted with the generally slower growth rate of $\xi_{c}$
with $T^{*}$ for $d\geq 2$, i.e.,

$(\xi_{c})^{d}\sim\frac{B}{A}\sim H^{-1},$ (95)

where we have used relation (93).

##  IX Conclusions and Discussion

We have derived a Fourier representation of the surface-area coefficient $B$
in terms of the structure factor $S({\bf k})$, which is especially useful when
scattering information is available experimentally or theoretically. In a
disordered system that is nearly hyperuniform, we showed that the ratio of
surface-area to volume coefficients, $B/A$, enables one to ascertain
hyperuniform and nonhyperuniform distance-scaling regimes of the variance
$\sigma^{2}(R)$ as a function of $R$ as well as the corresponding crossover
distance between these regimes. While the analysis of the ratio $B/A$ was
applied to systems that ultimately approached a class I hyperuniform state,
the corresponding extensions to other hyperuniformity classes is very
straightforward.

Using the ratio $B/A$, as well as other diagnostic measures of
hyperuniformity, including the hyperuniformity index $H$ and the direct-
correlation function length scale $\xi_{c}$, we characterized the structure of
three different exactly solvable models as a function of the relevant control
parameter, either density or temperature, with end states that are perfectly
hyperuniform. In the case of the sticky hard-sphere and stealthy models, we
studied the effect of dimensionality. For all three models, we showed that
these diagnostic hyperuniformity measures are positively correlated with one
another. Thus, the quantities $H$ and $\xi_{c}$ can also be employed to infer
the hyperuniform and nonhyperuniform distance-scaling regimes. This capacity
to determine hyperuniform scaling regimes is expected to be of great utility
in analyzing experimentally- or computationally-generated samples that are
necessarily of finite size.

In the same way that there is no perfect crystal, due to the inevitable
presence of imperfections, such as vacancies and dislocations, there is no
“perfect” hyperuniform system in laboratory practice, whether it is ordered or
not Kim and Torquato (2018). It is clear that the same diagnostic measures
explored in the present work can be used to detect the degree to which such
“imperfect” nearly hyperuniform systems deviate from perfect hyperuniformity.

We showed that the free-volume theory of the pressure of equilibrium packings
of identical hard spheres that approach a strictly jammed state, either along
the stable crystal or metastable disordered branch, dictates that such end
states be exactly hyperuniform. This implies that the packing on approach to
the jammed state must be ergodic in a generalized sense and hence free of any
defects, e.g., vacancies or dislocations along the crystal branch or rattlers
along the metastable fluid branch. It is an outstanding problem for future
research to place these implications of free-volume theory for the
hyperuniformity of strictly jammed packings on a firmer rigorous theoretical
foundation.

##  Appendix A Number Variance Scalings for Nonhyperuniform Systems

The local number variance $\sigma^{2}_{{}_{N}}(R)$ is generally a function
that can be decomposed into a global part that grows with the window radius
$R$ and a local part that oscillates on small length scales (e.g., mean
nearest-neighbor distance) about the global contribution. The more general
large-$R$ asymptotic formula for the variance is given by Torquato (2018)

$\sigma^{2}(R)=2^{d}\phi\Bigg{[}A_{{}_{N}}(R)\left(\frac{R}{D}\right)^{d}+B_{{}_{N}}(R)\left(\frac{R}{D}\right)^{d-1}+o\left(\frac{R}{D}\right)^{d-1}\Bigg{]},$
(96)

where the $R$-dependent volume and surface-area coefficients are respectively
given by

$A_{{}_{N}}(R)=1+\frac{\phi}{v_{1}(D/2)}\int_{|{\bf r}|\leq 2R}h({\bf r})d{\bf
r}$ (97)

and

$B_{{}_{N}}(R)=-\frac{\phi\,d\,\Gamma(d/2)}{2\,\pi^{1/2},D\,v_{1}(D/2)\Gamma[(d+1)/2]}\int_{|{\bf
r}|\leq 2R}h({\bf r})|{\bf r}|d{\bf r}.$ (98)

Observe that when the volume coefficient $A_{N}(R)$ and surface-area
coefficient $B_{N}(R)$ converge in the limit $R\rightarrow\infty$, they are
equal to the constants $A$ and $B$, defined by (24) and (25), respectively.
For this reason, $A$ and $B$ are called the global volume and surface-area
coefficients, respectively Torquato (2018).

An anti-hyperuniform system is one in which the exponent $\alpha$ [cf. (3)] is
negative $\alpha<0$, resulting in a structure factor that diverges in the
zero-wavenumber limit. The corresponding total correlation function $h({\bf
r})$ decays like $1/r^{d+\alpha}$ for large $r\equiv|\bf r|$. This power-law
tail controls the growth rate of $A_{{}_{N}}(R)$ with $R$; specifically,
carrying out the integration in (97) yields

$A_{{}_{N}}(R)\sim R^{-\alpha}.$ (99)

Substitution of this scaling behavior in the leading-order term of (96) gives
that the local number variance for an anti-hyperuniform system scales like
$\sigma^{2}(R)\sim R^{d-\alpha}$, which proves the anti-hyperuniform scaling
given in (8). Since $\alpha$ is negative, the number variance grows faster
than the window volume, i.e., faster than $R^{d}$. Up to a trivial constant in
the leading-order term in the asymptotic expansion of the number variance
given by (96), one can view a “typical” nonhyperuniform system in which $S(0)$
is bounded as one in which $\alpha=0^{+}$ (approaches zero from above), even
if $S(\bf k)$ is not described by a power-law form in the zero-wavenumber
limit. This enables one to conclude that $A_{{}_{N}}(R)$ converges to a
constant in the limit $R\rightarrow\infty$, thus yielding the $R^{d}$ scaling
behavior for $\sigma^{2}(R)$ in (8).

##  Appendix B Evaluation of the coefficients $A$ and $B$ for a Super-
Poissonian Point Pattern

Any nonhyperuniform point process for which $S(0)>1$ has a large-$R$
asymptotic number variance $\sigma^{2}(R)$ that is larger than that for a
Poisson point process [$S(0)=1$] with the same mean $\langle N(R)\rangle$ is
called super-Poissonian Torquato  _et al._ (2021). We evaluate the volume and
surface-area coefficients $A$ and $B$ for a specific model of a super-
Poissonian point process, namely, the Poisson cluster process, which is
characterized by strong clustering of the points with a large but finite value
of $S(0)$, and hence is far from being hyperuniform. The higher-order moments
of the number fluctuations as well as the corresponding probability
distribution for this model were recently studied Torquato  _et al._ (2021).
The construction of the cluster process starts from a homogeneous Poisson
point process of intensity $\rho_{p}$ Last and Penrose (2017). Each point of
the Poisson point process is the center of a cluster of points. The number of
points in each cluster is independent and follows a Poisson distribution with
mean value $c$. Following Ref. Torquato  _et al._ (2021), we consider here the
special case in which the positions of the points relative to the center of
the cluster follow an isotropic Gaussian distribution with standard deviation
$r_{0}$, which can be regarded to be the characteristic length scale of a
single cluster. In the infinite-volume limit, the structure factor for any $d$
is exactly given by Torquato  _et al._ (2021)

$S(k)=1+ce^{-k^{2}r_{0}^{2}}.$ (100)

It immediately follows from this formula and (24) that $S(0)=A=1+c$. Using the
Fourier representation of the surface-area coefficient (47) and relation
(100), we find that $B$, for any $d$, is exactly given by

$B=-\frac{c\,r_{0}\,d}{\sqrt{\pi}}.$ (101)

Remarkably, this model provides an example of a surface-area coefficient that
is negative (when $c$ is positive), which heretofore has not been identified.
Since the surface-area coefficient $B$ is derived from the large-$R$
asymptotic expansion (23) of the number variance $\sigma^{2}(R)$, which must
be positive, the length scale $r_{0}$ in (101), as any characteristic length
scale, must be much smaller than the window radius $R$.

##  Appendix C Sum Rules and Large-$k$ Asymptotic Behaviors of $S(k)$ for
Sphere Packings

Here we present sum rules as well as the exact large-$k$ asymptotic behaviors
of the structure factors of certain general packings of identical spheres that
apply in any space dimension $d$.

For a large class of packings of identical spheres of diameter $D$, the
following sum rule applies inside the hard core:

$\frac{r}{\left(2\pi
r\right)^{\frac{d}{2}}}\int_{0}^{\infty}k^{d/2}{\tilde{h}}(k)J_{\left(d/2\right)-1}(kr)\,dk=-1\qquad\mbox{for}\;\;r<D.$
(102)

This sum rule follows from the fact that $h({\bf r})=-1$ for $r<D$ and use of
(32). It is valid provided that the volume integral of ${\tilde{h}}({\bf k})$
over all reciprocal space is bounded. In the special case of $r=0$, (102)
yields

$\frac{d}{2^{d}\pi^{d/2}\Gamma(1+d/2)}\int_{0}^{\infty}k^{d-1}{\tilde{h}}(k)\,dk=-1.$
(103)

Such exact sum rules can be utilized to check the accuracy of the
determination of $S(k)$ of sphere packings via experimental or numerical
methods.

The Fourier transform of the indicator function $m(r;a)=\Theta(a-r)$ for a
$d$-dimensional sphere of radius $a$ is given by

$\displaystyle{\tilde{m}}(k;a)$ $\displaystyle=$
$\displaystyle\frac{(2\pi)^{d/2}}{k^{(d/2)-1}}\int_{0}^{a}r^{d/2}J_{(d/2)-1}(kr)dr$
(104) $\displaystyle=$
$\displaystyle\left(\frac{2\pi}{ka}\right)^{d/2}a^{d}J_{d/2}(ka).$

For large $k$, this Fourier transform is given by

${\tilde{m}}(k;a)\sim
R^{d}(2\pi)^{d/2}\sqrt{2/\pi}\;\frac{\cos[ka-(d+1)/4]}{(ka)^{(d+1)/2}}+{\cal
O}\left(\frac{1}{(ka)^{(d+3)/2}}\right)\qquad(ka\to\infty).$ (105)

For any packing of spheres of diameter $D$, $h(r)=-m(r;D)$ for $r<D$. Let us
assume that the contact-value of the pair correlation function, denoted by
$g_{2}(D^{+})$, is bounded, i.e., the jump discontinuity at contact is finite.
Using the result (105) for such a packing, we can relate the large-$k$
behavior of the structure factor in terms of $g_{2}(D^{+})$:

$\displaystyle S(k)$ $\displaystyle\sim
1-\phi\,g_{2}(D^{+})\,2^{3d/2}\Gamma(1+d/2)\sqrt{2/\pi}\;\frac{\cos[kD-(d+1)/4]}{(kD)^{(d+1)/2}}$
(106) $\displaystyle\hskip 25.29494pt+{\cal
O}\left(\frac{1}{(kD)^{(d+3)/2}}\right)\qquad(kD\to\infty).$

For $d=1,2$ and 3, the corresponding asymptotic behaviors are explicitly given
respectively by

$S(k)\sim 1-2\phi\,g_{2}(D^{+})\,\frac{\sin(kD)}{kD}+{\cal
O}\left(\frac{1}{(kD)^{2}}\right)\quad(d=1),$ (107) $S(k)\sim
1-8\sqrt{2/\pi}\,\phi\,g_{2}(D^{+})\,\frac{\cos(kD+\pi/4)}{(kD)^{3/2}}+{\cal
O}\left(\frac{1}{(kD)^{5/2}}\right)\quad(d=2),$ (108)

and

$S(k)\sim 1+24\phi\,g_{2}(D^{+})\,\frac{\cos(kD)}{(kD)^{2}}+{\cal
O}\left(\frac{1}{(kD)^{3}}\right)\quad(d=3).$ (109)

We see that the rate of decay of $S(k)$ increases as $d$ increases.

By contrast, if the sphere packing is disordered at a jammed state with
average contact number per particle of $Z$, then the jamming-contact condition
is described by

$g_{2}(r)\sim\frac{Z}{\rho s_{1}(1)D^{d-1}}\delta(r-D),$ (110)

where $s_{1}(1)=dv_{1}(1)$ is the $d$-dimensional surface area of a sphere of
unit radius. Thus, we see for such jammed packings, the large-$k$ behavior of
the structure factor is given by

$\displaystyle S(k)$ $\displaystyle\sim$ $\displaystyle
1+\frac{2^{d/2}\Gamma(1+d/2)\,Z}{d}\,\frac{J_{d/2-1}(kD)}{(kD)^{d/2-1}}$ (111)
$\displaystyle+{\cal
O}\left(\frac{1}{(kD)^{(d+1)/2}}\right)\qquad(kD\to\infty).$

For $d=1,2$ and 3, the corresponding asymptotic behaviors are explicitly given
respectively by

$S(k)\sim 1+Z\,\cos(kD)+{\cal O}\left(\frac{1}{(kD)}\right)\qquad(d=1)$ (112)
$S(k)\sim 1+Z\,J_{0}(kD)+{\cal O}\left(\frac{1}{(kD)^{3/2}}\right)\qquad(d=2)$
(113) $S(k)\sim 1+Z\;\frac{\sin(kD)}{kD}+{\cal
O}\left(\frac{1}{(kD)^{2}}\right)\qquad(d=3)\,.$ (114)

Again, we see that the rate of decay of $S(k)$ increases as $d$ increases, but
less rapidly than in unjammed packings at the same dimension in which the jump
discontinuity in $g_{2}(r)$ at contact is finite.

###### Acknowledgements.

The author is grateful to Jaeuk Kim, Michael Klatt, Charles Maher, Murray
Skolnick and Haina Wang for their careful reading of the manuscript. This work
was supported by the National Science Foundation under Award No. DGE-2039656.

## References

  * Torquato and Stillinger (2003) S. Torquato and F. H. Stillinger, “Local density fluctuations, hyperuniform systems, and order metrics,” Phys. Rev. E 68, 041113 (2003).
  * Torquato (2018) S. Torquato, “Hyperuniform states of matter,” Physics Reports 745, 1–95 (2018).
  * Hansen and McDonald (2013) J. P. Hansen and I. R. McDonald,  _Theory of Simple Liquids_ , 4th ed. (Academic Press, New York, 2013).
  * Florescu  _et al._ (2009) M. Florescu, S. Torquato, and P. J. Steinhardt, “Designer disordered materials with large complete photonic band gaps,” Proc. Nat. Acad. Sci. 106, 20658–20663 (2009).
  * De Rosa  _et al._ (2015) C. De Rosa, F. Auriemma, C. Diletto, R. Di Girolamo, A. Malafronte, P. Morvillo, G. Zito, G. Rusciano, G. Pesce, and A. Sasso, “Toward hyperuniform disordered plasmonic nanostructures for reproducible surface-enhanced Raman spectroscopy,” Phys. Chem. Chem. Phys. 17, 8061–8069 (2015).
  * Leseur  _et al._ (2016) O. Leseur, R. Pierrat, and R. Carminati, “High-density hyperuniform materials can be transparent,” Optica 3, 763–767 (2016).
  * Ma  _et al._ (2016) T. Ma, H. Guerboukha, M. Girard, A. D. Squires, R. A. Lewis, and M. Skorobogatiy, “3D printed hollow-core terahertz optical waveguides with hyperuniform disordered dielectric reflectors,” Adv. Optical Mater. 4, 2085–2094 (2016).
  * Zhang  _et al._ (2016a) G. Zhang, F. H. Stillinger, and S. Torquato, “Transport, geometrical and topological properties of stealthy disordered hyperuniform two-phase systems,” J. Chem. Phys 145, 244109 (2016a).
  * Gkantzounis  _et al._ (2017) G. Gkantzounis, T. Amoah, and M. Florescu, “Hyperuniform disordered phononic structures,” Phys. Rev. B 95, 094120 (2017).
  * Froufe-Pérez  _et al._ (2017) L. S. Froufe-Pérez, M. Engel, J. José Sáenz, and F. Scheffold, “Transport Phase Diagram and Anderson Localization in Hyperuniform Disordered Photonic Materials,” Proc. Nat. Acad. Sci. 114, 9570––9574 (2017).
  * Chen and Torquato (2018) D. Chen and S. Torquato, “Designing disordered hyperuniform two-phase materials with novel physical properties,” Acta Materialia 142, 152–161 (2018).
  * Zhang  _et al._ (2018) H. Zhang, W. Wu, and Y. Hao, “Luneburg lens from hyperuniform disordered composite materials,” in  _2018 IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting_ (2018) pp. 2281–2282.
  * Gorsky  _et al._ (2019) S. Gorsky, W. A. Britton, Y. Chen, J. Montaner, A. Lenef, M. Raukas, and L. Dal Negro, “Engineered hyperuniformity for directional light extraction,” APL Photonics 4, 110801 (2019).
  * Sheremet  _et al._ (2020) A. Sheremet, R. Pierrat, and R. Carminati, “Absorption of scalar waves in correlated disordered media and its maximization using stealth hyperuniformity,” Phys. Rev. A 101, 053829 (2020).
  * Kim and Torquato (2020) J. Kim and S. Torquato, “Multifunctional composites for elastic and electromagnetic wave propagation,” Proc. Nat. Acad. Sci. 117, 8764–8774 (2020).
  * Yu  _et al._ (2021) S. Yu, C.-W. Qiu, Y. Chong, S. Torquato, and N. Park, “Engineered disorder in photonics,” Nature Rev. Mater. 6, 226–243 (2021).
  * Zachary and Torquato (2009) C. E. Zachary and S. Torquato, “Hyperuniformity in point patterns and two-phase heterogeneous media,” J. Stat. Mech.: Theory & Exp. 2009, P12015 (2009).
  * Lin  _et al._ (2017) C. Lin, P. J. Steinhardt, and S. Torquato, “Hyperuniformity variation with quasicrystal local isomorphism class,” J. Phys.: Cond. Matter 29, 204003 (2017).
  * Oğuz  _et al._ (2017) E. C. Oğuz, J. E. S. Socolar, P. J. Steinhardt, and S. Torquato, “Hyperuniformity of quasicrystals,” Phys. Rev. B 95, 054119 (2017).
  * Gabrielli (2004) A. Gabrielli, “Point processes and stochastic displacement fields,” Phys. Rev. E 70, 066131 (2004).
  * Gabrielli and Torquato (2004) A. Gabrielli and S. Torquato, “Voronoi and void statistics for superhomogeneous point processes,” Phys. Rev. E 70, 041105 (2004).
  * Gabrielli  _et al._ (2008) A. Gabrielli, M. Joyce, and S. Torquato, “Tilings of space and superhomogeneous point processes,” Phys. Rev. E 77, 031125 (2008).
  * Kim and Torquato (2018) J. Kim and S. Torquato, “Effect of imperfections on the hyperuniformity of many-body systems,” Phys. Rev. B 97, 054105 (2018).
  * Uche  _et al._ (2004) O. U. Uche, F. H. Stillinger, and S. Torquato, “Constraints on collective density variables: Two dimensions,” Phys. Rev. E 70, 046122 (2004).
  * Torquato  _et al._ (2015) S. Torquato, G. Zhang, and F. H. Stillinger, “Ensemble theory for stealthy hyperuniform disordered ground states,” Phys. Rev. X 5, 021020 (2015).
  * Zhang  _et al._ (2016b) G. Zhang, F. H. Stillinger, and S. Torquato, “The perfect glass paradigm: Disordered hyperuniform glasses down to absolute zero,” Sci. Rep. 6, 36963 (2016b).
  * Lei and Ni (2019) Q-L. Lei and R. Ni, “Hydrodynamics of random-organizing hyperuniform fluids,” Proc. Nat. Acad. Sci. 116, 22983–22989 (2019).
  * Torquato  _et al._ (2019) S. Torquato, G. Zhang, and M. de Courcy-Ireland, “Hidden multiscale order in the primes,” J. Phys. A: Math. & Theoretical 52, 135002 (2019).
  * Donev  _et al._ (2005a) A. Donev, F. H. Stillinger, and S. Torquato, “Unexpected density fluctuations in disordered jammed hard-sphere packings,” Phys. Rev. Lett. 95, 090604 (2005a).
  * Zachary  _et al._ (2011) C. E. Zachary, Y. Jiao, and S. Torquato, “Hyperuniform long-range correlations are a signature of disordered jammed hard-particle packings,” Phys. Rev. Lett. 106, 178001 (2011).
  * Jiao and Torquato (2011) Y. Jiao and S. Torquato, “Maximally random jammed packings of Platonic solids: Hyperuniform long-range correlations and isostaticity,” Phys. Rev. E 84, 041309 (2011).
  * Atkinson  _et al._ (2016) S. Atkinson, G. Zhang, A. B. Hopkins, and S. Torquato, “Critical slowing down and hyperuniformity on approach to jamming,” Phys. Rev. E 94, 012902 (2016).
  * Feynman and Cohen (1956) R. P. Feynman and M. Cohen, “Energy spectrum of the excitations in liquid helium,” Phys. Rev. 102, 1189–1204 (1956).
  * Reatto and Chester (1967) L. Reatto and G. V. Chester, “Phonons and the properties of a Bose system,” Phys. Rev. 155, 88–100 (1967).
  * Torquato  _et al._ (2008) S. Torquato, A. Scardicchio, and C. E. Zachary, “Point processes in arbitrary dimension from Fermionic gases, random matrix theory, and number theory,” J. Stat. Mech.: Theory Exp. 2008, P11019 (2008).
  * Zachary and Torquato (2011) C. E. Zachary and S. Torquato, “Anomalous local coordination, density fluctuations, and void statistics in disordered hyperuniform many-particle ground states,” Phys. Rev. E 83, 051133 (2011).
  * Hexner and Levine (2015) D. Hexner and D. Levine, “Hyperuniformity of critical absorbing states,” Phys. Rev. Lett. 114, 110602 (2015).
  * Widom (1965) B. Widom, “Equation of state in the neighborhood of the critical point,” J. Chem. Phys. 43, 3898–3905 (1965).
  * Kadanoff (1966) L. P. Kadanoff, “Scaling laws for Ising models near $T_{c}$,” Physics 2, 263–272 (1966).
  * Fisher (1967) M. E. Fisher, “The theory of equilibrium critical phenomena,” Rep. Prog. Phys. 30, 615 (1967).
  * Wilson and Kogut (1974) K. G. Wilson and J. Kogut, “The renormalization group and the $\epsilon$ expansion,” Phys. Rep. 12, 75–199 (1974).
  * Binney  _et al._ (1992) J. J. Binney, N. J. Dowrick, A. J. Fisher, and M. E. J. Newman,  _The Theory of Critical Phenomena: An Introduction to the Renormalization Group_ (Oxford University Press, Oxford, England, 1992).
  * Oğuz  _et al._ (2019) E. C. Oğuz, J. E. S. Socolar, P. J. Steinhardt, and S. Torquato, “Hyperuniformity and anti-hyperuniformity in one-dimensional substitution tilings,” Acta Cryst. Section A: Foundations & Advances 75 (2019).
  * Dreyfus  _et al._ (2015) R. Dreyfus, Y. Xu, T. Still, L. A. Hough, A. G. Yodh, and S. Torquato, “Diagnosing hyperuniformity in two-dimensional, disordered, jammed packings of soft spheres,” Phys. Rev. E 91, 012302 (2015).
  * Ornstein and Zernike (1914) L. S. Ornstein and F. Zernike, “Accidental deviations of density and opalescence at the critical point of a single substance,” Proc. Akad. Sci. (Amsterdam) 17, 793–806 (1914).
  * Hopkins  _et al._ (2012) A. B. Hopkins, F. H. Stillinger, and S. Torquato, “Nonequilibrium static diverging length scales on approaching a prototypical model glassy state,” Phys. Rev. E 86, 021505 (2012).
  * Torquato  _et al._ (2000) S. Torquato, T. M. Truskett, and P. G. Debenedetti, “Is random close packing of spheres well defined?” Phys. Rev. Lett. 84, 2064–2067 (2000).
  * Marcotte  _et al._ (2013) É. Marcotte, F. H. Stillinger, and S. Torquato, “Nonequilibrium static growing length scales in supercooled liquids on approaching the glass transition,” J. Chem. Phys. 138, 12A508 (2013).
  * Chremos and Douglas (2017) A. Chremos and J. F. Douglas, “Particle localization and hyperuniformity of polymer-grafted nanoparticle materials,” Annalen der Physik 529 (2017).
  * Chremos and Douglas (2018) A. Chremos and J. F Douglas, “Hidden hyperuniformity in soft polymeric materials,” Phys. Rev. Lett. 121, 258002 (2018).
  * Martelli  _et al._ (2017) F. Martelli, S. Torquato, N. Giovambattista, and R. Car, “Large-scale structure and hyperuniformity of amorphous ices,” Phys. Rev. Lett. 119, 136002 (2017).
  * Zhou  _et al._ (2020) Y. Zhou, B. Mei, and K. S. Schweizer, “Integral equation theory of thermodynamics, pair structure, and growing static length scale in metastable hard sphere and weeks-chandler-andersen fluids,” Phys. Rev. E 101, 042121 (2020).
  * Klatt  _et al._ (2019) M. A. Klatt, J. Lovrić, D. Chen, S. C. Kapfer, F. M. Schaller, P. W. A. Schönhöfer, B. S. Gardiner, A-S. Smith, G. E. Schröder-Turk, and S. Torquato, “Universal hidden order in amorphous cellular geometries,” Nature Comm. 10, 811 (2019).
  * Hain  _et al._ (2020) T. M. Hain, M. A. Klatt, and G. E. Schröder-Turk, “Low-temperature statistical mechanics of the Quantizer problem: Fast quenching and equilibrium cooling of the three-dimensional Voronoi liquid,” J. Chem. Phys. 153, 234505 (2020).
  * Zhang and Torquato (2020) G. Zhang and S. Torquato, “Realizable hyperuniform and nonhyperuniform particle configurations with targeted spectral functions via effective pair interactions,” Phys. Rev. E 101, 032124 (2020).
  * Torquato  _et al._ (2021) S Torquato, J. Kim, and M. A. Klatt, “Local number fluctuations in hyperuniform and nonhyperuniform systems: Higher-order moments and distribution functions,” Phys. Rev. X (2021).
  * Torquato (2002) S. Torquato,  _Random Heterogeneous Materials: Microstructure and Macroscopic Properties_ (Springer-Verlag, New York, 2002).
  * Alder and Wainwright (1957) B. J. Alder and T. E. Wainwright, “Phase transition for a hard sphere system,” J. Chem. Phys. 27, 1208–1209 (1957).
  * Frenkel (1999) D. Frenkel, “Entropy-driven phase transitions,” Physica A 263, 26–38 (1999).
  * Mau and Huse (1999) S. C. Mau and D. A. Huse, “Stacking entropy of hard-sphere crystals,” Phys. Rev. E 59, 4396–4401 (1999).
  * Torquato and Stillinger (2010) S. Torquato and F. H. Stillinger, “Jammed hard-particle packings: From Kepler to Bernal and beyond,” Rev. Mod. Phys. 82, 2633 (2010).
  * Torquato and Stillinger (2001) S. Torquato and F. H. Stillinger, “Multiplicity of generation, selection, and classification procedures for jammed hard-particle packings,” J. Phys. Chem. B 105, 11849–11853 (2001).
  * Donev  _et al._ (2004) A. Donev, S. Torquato, F. H. Stillinger, and R. Connelly, “A linear programming algorithm to test for jamming in hard-sphere packings,” J. Comput. Phys. 197, 139–166 (2004).
  * Torquato and Stillinger (2007) S. Torquato and F. H. Stillinger, “Toward the jamming threshold of sphere packings: Tunneled crystals,” J. Appl. Phys. 102, 093511 (2007), Erratum, 103, 129902 (2008).
  * Torquato  _et al._ (2003) S. Torquato, A. Donev, and F. H. Stillinger, “Breakdown of elasticity theory for jammed hard-particle packings: Conical nonlinear constitutive theory,” Int. J. Solids Structures 40, 7143–7153 (2003).
  * Salsburg and Wood (1962) Z. W. Salsburg and W. W. Wood, “Equation of state of classical hard spheres at high density,” J. Chem. Phys. 37, 798–1025 (1962).
  * Stillinger and Salsburg (1969) F. H. Stillinger and Z. W. Salsburg, “Limiting polytope geometry for rigid rods, disks, and spheres,” J. Stat. Phys. 1, 179–225 (1969).
  * Donev  _et al._ (2005b) A. Donev, S. Torquato, and F. H. Stillinger, “Pair correlation function characteristics of nearly jammed disordered and ordered hard-sphere packings,” Phys. Rev. E 71, 011105: 1–14 (2005b).
  * Parisi and Zamponi (2010) G. Parisi and F. Zamponi, “Mean field theory of hard sphere glasses and jamming,” Rev. Mod. Phys. 82, 789—845 (2010).
  * Percus (1964) J. Percus, “Pair distribution function in classical statistical mechanics,” in  _The Equilibrium Theory of Classical Fluids_ , edited by H. L. Frisch and J. L. Lebowitz (Benjamin, 1964).
  * Tonks (1936) L. Tonks, “The complete equation of state of one, two and three-dimensional gases of hard elastic spheres,” Phys. Rev. 50, 955–963 (1936).
  * Kim and Torquato (2017) J. Kim and S. Torquato, “Effect of window shape on the detection of hyperuniformity via the local number variance,” J. Stat. Mech.: Th. and Exper. 2017, 013402 (2017).
  * Torquato and Stillinger (2002) S. Torquato and F. H. Stillinger, “Controlling the short-range order and packing densities of many-particle systems,” J. Phys. Chem. B 106, 8354–8359 (2002), Erratum 106, 11406 (2002).
  * Cohn and Elkies (2003) H. Cohn and N. Elkies, “New upper bounds on sphere packings I,” Annals Math. 157, 689–714 (2003).
  * Torquato and Stillinger (2006) S. Torquato and F. H. Stillinger, “New conjectural lower bounds on the optimal density of sphere packings,” Experimental Math. 15, 307–331 (2006).
  * Uche  _et al._ (2006) O. U. Uche, F. H. Stillinger, and S. Torquato, “On the realizability of pair correlation functions,” Physica A 360, 21–36 (2006).
  * Ball (1992) K. Ball, “A lower bound for the optimal density of lattice packings,” Int. Math. Res. Notices 68, 217–221 (1992).
  * Batten  _et al._ (2008) R. D. Batten, F. H. Stillinger, and S. Torquato, “Classical disordered ground states: Super-ideal gases, and stealth and equi-luminous materials,” J. Appl. Phys. 104, 033504 (2008).
  * Last and Penrose (2017) G. Last and M. Penrose,  _Lectures on the Poisson Process_, Institute of Mathematical Statistics Textbooks (Cambridge University Press, Cambridge, United Kingdom, 2017).