Volume 17
Number 39
21 October 2021
Pages 8761–8984

Soft Matterrsc.li/soft-matter-journal

ISSN 1744-6848

PAPER
Hiroshi Frusawa
Non-hyperuniform metastable states around a disordered
hyperuniform state of densely packed spheres: stochastic
density functional theory at strong coupling

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Non-hyperuniform metastable states arounda disordered hyperuniform state of densely
packed spheres: stochastic density functional
theory at strong coupling
Hiroshi Frusawa
The disordered and hyperuniform structures of densely packed spheres near and at jamming arecharacterized by vanishing of long-wavelength density fluctuations, or equivalently by long-range
power-law decay of the direct correlation function (DCF). We focus on previous simulation results thatexhibit the degradation of hyperuniformity in jammed structures while maintaining the long-range
nature of the DCF to a certain length scale. Here we demonstrate that the field-theoretic formulation ofstochastic density functional theory is relevant to explore the degradation mechanism. The strongcoupling expansion method of stochastic density functional theory is developed to obtain themetastable chemical potential considering the intermittent fluctuations in dense packings. The
metastable chemical potential yields the analytical form of the metastable DCF that has a short-range

Received 16th July 2021,
Accepted 5th September 2021

cutoﬀ inside the sphere while retaining the long-range power-law behavior. It is confirmed that the

DOI: 10.1039/d1sm01052b

with the previous simulation results of degraded hyperuniformity. We can also predict the emergence ofmetastable DCF provides the zero-wavevector limit of the structure factor in quantitative agreementsoft modes localized at the particle scale by plugging this metastable DCF into the linearized Dean–rsc.li/soft-matter-journal

Kawasaki equation, a stochastic density functional equation.

I. Introduction
Hyperuniformity is characterized by density fluctuations that
decrease to zero at the longest scales.1–17 We have observed thehyperuniform states in a variety of complex soft matter systems,
including foams, polymer blends, colloidal suspensions andbiological tissues (see ref. 1 and 2 for reviews). It has also beenfound that non-crystalline materials with hyperuniformity haveunique physical properties such as high-density transparency
and isotropic filtration of elastic or electromagnetic waves.1–4
Consequently, considerable attention has been given to disordered hyperuniform materials fabricated at the micro- andnano-scales, because of their potential importance for applications in photonics, electronics, and structural components withnovel properties.3,4Here we focus on computer glasses among the disordered
hyperuniform systems. Recent methodological developmentsallow us to create computer glasses in an experimentally
relevant regime,1–43 and yet the disordered hyperuniformity at
jamming has not always been realized.1,2,38–43 The emergenceof hyperuniformity depends on the preparation protocols,Laboratory of Statistical Physics, Kochi University of Technology, Tosa-Yamada,Kochi 782-8502, Japan. E-mail: frusawa.hiroshi@kochi-tech.ac.jp8810 | Soft Matter, 2021, 17, 8810–8831

partly because of the significantly long computational timethat is required to determine the configurations near and at
jamming.1,2,5–14
On the one hand, some simulation studies have demonstrated the hyperuniformity in densely packed spheres: thestructure factor S(k) in a hyperuniform state exhibits a nontrivial linear dependence on the wavevector magnitude k in the
low-wavevector range near and at jamming (i.e., S(k) B k (k Z0)), and the zero-wavevector limit of the structure factor
S(0) eventually vanishes at jamming.1,2,5–14 These results indicate not only the existence of long-range order, but also
the complete suppression of density fluctuations over thesystem scale.
Meanwhile, other simulation studies near and at
jamming38–43 provide the non-vanishing structure factor at
the zero wavevector. The degradation of hyperuniformity is thateither saturation or an upturn is observed for S(k) at the lowest
values of k, despite the linear relation above the crossover
wavevector kc:38–43SðkÞ  k ðk  kc Þ;

lim SðkÞ  Sðkc Þ:

k!0

(1)

It has also been demonstrated that S(0) is weakly dependent on
the density variation.38–43

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The quantitative diﬀerence between the non-hyperuniform
and hyperuniform states can be seen from the inverse of thezero-wavevector structure factors. While the non-vanishingvalues of S(0) due to the incomplete linear-dependence ofS(k) are in the range of38–43
102 o

1o 103 ;
Sð0Þ(2)

the hyperuniform computer glasses have been characterized byinequality,1
4 104 ;
Sð0Þ

(3)

irrespective of system details.1,2,5–14
In terms of density–density correlation functions in real
space, hyperuniformity is a kind of inverted critical phenomenon. It is among the critical phenomena in normal fluids that
total correlation functions are long-ranged at critical points,
accompanied by the diverging behaviors of density fluctuations
and isothermal compressibility, while keeping the direct correlation function (DCF) short-ranged. In contrast, the inverted
critical phenomenon is that the hyperuniform DCF is longranged in correspondence with the vanishing isothermal compressibility, despite the absence of long-range behavior for the
total correlation function.1,2,5–14,17
More concretely, the long-range behavior of the hyperuniform DCF c(r) is described by the power-law as follows:1,2,5–
7,13,171cðrÞ  2r

ðr 4 sÞ;(4)

where |r| = r and s denotes the sphere diameter. The long-range
decay of c(r) reads c(k) B k1 in the Fourier space, which isequivalent to the linear behavior S(k) B k due to the following
relation between S(k) and c(k):1¼ 1  ncðkÞ;
SðkÞ

(5)

with n being the spatially averaged density of spheres. Furthermore, in disordered packings of hard spheres, the hyperuniform DCF for r r s satisfies another power-law,13cðrÞ 

1ðr  sÞ;r(6)

which is divergent at small r.
It follows from eqn (1) and (5) that the non-hyperuniform
DCF c(r) at jamming satisfies the scaling relation (4) over afinite range. In other words, the violation of hyperuniformity
occurs while maintaining the long-range behavior to a lengthscale Lc: eqn (4) holds in the range of s o r r Lc38–43 where
6

Lc
 10:
s

Table 1 Four types of hyperuniform and non-hyperuniform systems. We
investigate the type N1 of non-hyperuniform systems (see Sec. IV for thedetails)

Characterization of the DCF(7)

It is to be noted that the simulation results of hyperuniform
systems are also likely to provide the finiteness represented byeqn (7),1,2,5–14 for we have computational difficulty obtaining
the scaling behavior (c(r) B r2) over Lc from the structureThis journal is © The Royal Society of Chemistry 2021

factor, irrespective of whether the computer glass is in a
hyperuniform or non-hyperuniform state.
This common feature of the long-range behavior (eqn (4)) in
the hyperuniform and non-hyperuniform DCFs raises the question of what causes the diﬀerence between eqn (2) and (3).
Accordingly, it is the purpose of this study to reveal the underlying mechanism behind the diﬀerence between the emergenceand degradation of hyperuniformity. To this end, we formulate
an analytical form of the non-vanishing zero-wavevector
structure factor that satisfies eqn (2) under the condition ofeqn (7). A key ingredient of our formulation is the strongcoupling approximation of stochastic density functional theory(DFT)44–56 which can consider intermittent fluctuations while
fixing the density field at a given distribution of dense packingsnear and at jamming.
The remainder of this paper consists of two parts. In theformer part of Sections II–IV, the problem to be addressed isdefined. Section II provides the theoretical background as towhy stochastic DFT should be brought into the problem offluctuation-induced non-hyperuniformity. In Section III, thebasic formulation of stochastic DFT is presented, focusing on
the definition of metastable states. Then, we find that stochastic DFT allows us to relate the metastable zero-wavevector
structure factor S*(0) to the potential energy l* per particle,
which will be referred to as the metastable chemical potential.
In Section IV, the non-hyperuniform state on the target is
specified using Table 1, which shows the classification list of
hyperuniform and non-hyperuniform systems.
We see from the system specification that the nonhyperuniformity of our concern requires the short-range cutoﬀ
of the metastable DCF c*(r), as well as a drop in the long-ranged
DCF for r 4 Lc. Our primary goal is to derive the short-range
cutoﬀ of the DCF by developing the strong-coupling approximation for stochastic DFT.
The latter part of this paper presents the results anddiscussion regarding the metastable DCF c*(r). Before entering
the main results, Section V compares the stochastic DFT withthe equilibrium DFT57–74 in terms of S*(0). It is demonstrated
as a preliminary result that the resulting forms of S*(0) in theequilibrium and stochastic DFTs coincide with each other as
far as the Gaussian approximation of stochastic DFT isadopted. Section VI provides the metastable DCF expressedby the Mayer-type function form, hence verifying the cutoff forthe metastable DCF c*(r) inside the sphere. As a consequence,

SystemPower-law decay

Magnitude at zeroseparation

Type

Hyperuniform

Complete
Incomplete
Incomplete
Absent—Divergent
FiniteFinite

H1
H2
N1
N2

Non-hyperuniform

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Table 2Paper

Comparison between the theoretical approaches and results (see also Sec. VIIIC). We follow the notation of the type names given in Table 1

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State description

Statics

DFT type

ApproximationTypem or l*DCF

Equation

Short-range

Long-range

Equilibrium
Stochastic
Stochastic

Ramakrishnan–Yussouff
Gaussian
Strong-coupling

H2
H2
N1

Eqn (31)
Eqn (47)
Eqn (51)

Eqn (44) or (56)
Eqn (44) or (56)
Eqn (52) or (54)

Eqn (29)
Eqn (14) or (86)Eqn (14) or (86)

FrozenFrozenSoft

Correlated
Correlated
Correlated

we confirm that S*(0) satisfies relation (2), instead of eqn (3). In
Section VII, the coupling constant g to represent the strength ofinteractions is introduced using the hyperuniform DCF c(r),and it is shown that the 1/g expansion method becomes
equivalent to the virial-type one at the strong coupling of g c
1. Correspondingly, the interaction term in the metastablechemical potential l* is expressed by the above metastable
DCF c*(r). In Section VIII, the stochastic density functional
equation clarifies that the short-range cutoff of the metastable
DCF c*(r) leads to the emergence of dynamic softening at theparticle scale: the interaction-induced restoring force against
density fluctuations around a metastable state vanishes within
the scale of spherical diameter s. The microscopic mechanism
behind the soft modes is also discussed in connection
with previous simulation results. Furthermore, both Fig. 3
and Table 2 summarize the static and dynamic results forcomparing equilibrium DFT, stochastic DFT in the Gaussian
approximation, and stochastic DFT in the strong-coupling
approximation. The final remarks are given in Section IX.

II. A theoretical background of
stochastic DFTThis section is intended to provide a brief overview of theoretical
approaches to jammed structures for explaining the relevance ofstochastic DFT44–56 to the degradation of hyperuniformity.
A.

Marginal stability and the free energy landscape

There have been two conceptual approaches to address various
issues on computer glasses, including the structure factors thatvary depending on the protocols used.1,2,5–14,16 One is theensemble approach to investigate physically relevant packings
based on the packing protocol selected. The other method is
the geometric-structure approach for the quantitative characterization of single-packing configurations to enumerate andclassify the jammed structures.
On the one hand, the ensemble approach has involved the
problem that the protocol-dependency of the occurrence frequency of jammed configurations leads to the ambiguity ofweighing jammed states.1,2,5–17 Recently, however, the protocoldependency problem is theoretically tackled: the canonicalensemble method is developed for a large number of allowed
configurations to resolve the configuration realizability issue.16
The geometric-structure study, on the other hand, has
distinguished three types of jamming for densely packed
spheres:1,2,11 local, collective and strict jammings. These typesof jamming are hierarchical in that local and collective

8812 | Soft Matter, 2021, 17, 8810–8831

Dynamics

jammings are prerequisites for collective and strict ones,
respectively, as follows: (i) in locally jammed states, a particle
cannot translate when the positions of all other spheres in thepacking are fixed; (ii) in collectively jammed states, particles
prevented from translating are further stable to uniform compression; (iii) strictly jammed packings are stable against both
uniform and shear deformations.
The geometric-structure studies on various computer glasses
have verified that the hyperuniformity emerges in either strictly
or collectively jammed systems having isostaticity.1,2,8 Here theisostatic configuration provides a mean contact number 2d per
particle with d being the spatial dimension, thereby enhancing
the mechanical stability. To be noted, however, the mechanicalrigidity of jammed packings is a necessary but not suﬃcientcondition for hyperuniformity.
It has been conjectured that any strictly jammed saturated
infinite packing of identical spheres is hyperuniform; theconjecture excludes the existence of rattlers or particles thatare free to move in a confining cage, by definition of strictly
jammed packings.1,2,5,12,17 Conversely, it depends on simulation methods and conditions whether dense packings otherthan the strictly jammed ones, including the isostatic andcollectively jammed systems, are hyperuniform or not. The
isostatic systems can be destabilized by cutting one particle
contact unless disordered packings are strictly jammed. In
other words, isostaticity is a critical factor in mechanicalmarginal stability.1,2,12,18–37,75–83
Recent simulation results have demonstrated that thermal
fluctuations in the marginal states are accompanied by theintermittent rearrangements of particles.18–37 As a consequence, the marginal systems become responsive to have lowfrequency soft modes that are nonphononic and anharmonic.
For instance, quasi-localized modes coupled to an elastic
matrix create soft spots composed of tens to hundreds ofparticles undergoing displacements.18–37 The low-frequency
soft modes exhibit similar behaviors, and the common featuresof marginal states have been related to the emergence of many
local minima in the free-energy landscape.26,84–87
The similarity in anharmonic vibrations suggests that theensemble of configurations visited by the slow dynamics could
reveal the characteristics of marginal stability associated withthe free-energy landscape, even though possible configurations
depend on a protocol adopted.1,2,5–14,16
B. The free-energy density functional: comparison between
stochastic and equilibrium DFTs
For assessing the applicability of density functional approaches
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stochastic DFT44–56 and equilibrium DFT, or classical DFT
conventionally used.57–74
Equilibrium DFT,57–74 one of the ensemble approaches,
has been found relevant to investigate the free-energy
landscape.84–91 It has been demonstrated that metastable minima
determined by equilibrium DFT are not only correlated with theappearance of two-step relaxation and divergence of relaxation
time, but are also directly connected with dynamical
heterogeneity.62–74 In equilibrium DFT, the metastable density
profile r*(r) has been approximated by the superposition of narrow
Gaussian density profiles centered around a set of points forming
an aperiodic lattice. Equilibrium DFT has properly identified themetastable state of a liquid having an inhomogeneous and aperiodic density as a local minimum of the equilibrium free-energyfunctional with respect to the variation in the width parameter forthe above-mentioned Gaussian distribution.62–74
However, the violation of perfect hyperuniformity has been
beyond the scope of equilibrium DFT. Recently, the following
three scenarios of imperfections have been proposed fordemonstrating the degradation of hyperuniformity both theoretically and numerically:15 (i) uncorrelated point defects, (ii)stochastic particle displacements that are spatially correlated,and (iii) thermal excitations. In this study, we focus on thesecond scenario (ii) that is related to intermittent particle
rearrangements in a contact network, a set of bonds connecting
particles which are in contact with each other (see ref. 18–22, 75
and 76 for reviews). The elastic nature of the contact network
could be responsible for the above-mentioned second scenario
of non-hyperuniformity, or the spatially correlated displacements occurring stochastically; this will be discussed in Sections VIII and IX, based on the results obtained herein.
From stochastic DFT,44–56 on the other hand, it is expectedthat the above-mentioned second scenario (i.e., (ii) stochastic
and spatially correlated displacements) could be described interms of stochastic density dynamics. To see this, a brief review
of stochastic DFT is given below.
Stochastic DFT has been used as one of the most powerfultools for describing slowly fluctuating and/or intermittent
phenomena, such as glassy dynamics, nucleation or patternformation of colloidal particles, dielectric relaxation of Brownian dipoles, and even tumor growth (see ref. 44 for a thoroughreview). The stochastic density functional equation, which hasoften been referred to as the Dean–Kawasaki equation,44,45
forms the basis of stochastic DFT. It has been shown in various
systems that the Dean–Kawasaki equation successfully
describes the stochastic evolution of the instantaneous microscopic density field of overdamped Brownian particles. Of greatpractical use is the Dean–Kawasaki equation linearized withrespect to density fluctuations around various reference density
distributions.44,50–52,54–56
As seen below, stochastic DFT is formulated on the hybrid
framework that combines equilibrium DFT and statistical field
theory.54,55,92 The hybrid framework allows us to investigate themetastable states considering fluctuations as clarified below. In
Section VIII, stochastic DFT will also shed light on the dynamical properties of non-hyperuniformity.

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III. Basic formulation: stochastic DFTThis section shows that stochastic DFT44–56 is available toinvestigate the zero-wavevector limit of the structure factor
S*(0) in a metastable state. It is not merely a review of theprevious formulations,54 but rather a revisit for making it clearthat stochastic DFT serves as a systematic evaluation of S*(0): as
demonstrated in Sections VI and VII, we can evaluate the extent
to which S*(0) is altered to the stochastic fluctuations around a
hyperuniform state in a systematic manner.
First, the constrained free-energy functional A½r is represented by the hybrid form using the functional and configurational integrals (Section IIIA). Next, we introduce the nonequilibrium excess chemical potential appearing in the stochastic
DFT equation (Section IIIB). Third, the metastable state is defined
based on stochastic DFT (Section IIIC). Last, the metastable DCF
c*(r) is generated from the metastable chemical potential l*,thereby yielding S*(0) expressed by c*(r) (Section IIID).
A. Constrained free-energy functional A½r in connection with
the Fokker–Planck equation^ N ðr; tÞ ¼
Let r

N
P

d½r  ri ðtÞ be the instantaneous microscopici¼1

density of an N-particle system where the position ri(t) at time t
represents an instantaneous location of the i-th particle. The
corresponding distribution functional P[r, t] of the density fieldr(r, t) is defined by*+Y^N ðr; tÞ  rðr; tÞ ;P½r; t ¼
d½r
(8)
r

where hOi signifies the noise-averaging operation for O in theoverdamped dynamics.
As detailed in Appendix A, P[r, t] satisfies the Fokker–Planck
equation given by eqn (A1). It follows from the stationary condition qPst[r]/qt = 0 on the Fokker–Planck equation that the distribution functional in a steady state, Pst[r], is determined by thefree-energy functional A½r of a given density field r(r, t):
Pst ½r ¼ ÐeA½r
:DreA½r

(9)

We can evaluate the constrained free-energy functional A½r byintroducing a fluctuating potential field f as follows (see Appendix A for the derivation of eqn (10)–(13)):
ð
A½r
e
¼ DfeF½r;f D½r;(10)
where D[r] denotes the constraint due to the canonical ensemble:
ð

8
>
>
drrðrÞ ¼ N<1ð
D½r ¼(11)>>:0drrðrÞaN :The functional F[r, f] in the exponent of eqn (10) is defined using
the grand potential as follows:

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Ð
Y
Y
eF½r;fþ drrðrÞm ¼ Tremþifðri Þcdft ðri Þevðri rj ÞOpen Access Article. Published on 23 September 2021. Downloaded on 10/19/2024 11:08:25 PM.
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i

i;j

Ð
 e drrðrÞfcdft ðrÞifðrÞg
Ð¼ eOfcdft ifgþ drrðrÞfcdft ðrÞifðrÞg ;

(12)

1 1 Ð
Ð
P
dr1
drN , m denotes the equilibrium
N¼0 N!
chemical potential, v(r) is the original interaction potential including the hard sphere potential and Lennard-Jones potential, andO[c] is the grand potential in the presence of an external field c(r).
Here it is noted that all the energetic quantities used in this study(e.g., m, v(r) and O[c]) are given in kBT-unit.
As clearly seen from Appendix A, the fluctuating potential
field f(r) in eqn (12) traces back to the auxiliary field for the
Fourier transform of the Dirac delta functionalQ^N ðr; tÞ  rðr; tÞ.54,55,92 Also, the functional F[r, f  0] in
d½r

Ð
number constraint drrðrÞ ¼ N and is reduced to thechemical potential m when considering equilibrium DFT (seeSection IIIC).
In the Gaussian approximation of the f-field, we havewhere Tr rthe absence of the f-field corresponds to the intrinsic Helmholtz free energy in the presence of the external field cdft(r).
Therefore, the following relation holds:
dF½r; 0
¼ m  cdft ðrÞ;drðrÞwðr  r0 Þ ¼

(14)

(15)

(16)

dF½r; 0 dDFdft ½r; fþ lN ;
drdr

dF½r; 0 dDFdft ½r; fþ lN ;
drdr(23)

dF½r; fdF½r; 0dDFdft ½r; frlex ½r ¼ r
¼r
þr
dr
dr
dr

(24)

because |rlN| = 0. In Section V, we will evaluate the second
term on the right-hand side (rhs) of eqn (23), dDFdft ½r; f=dr,following the above-mentioned Gaussian approximation,
whereas the strong-coupling approximation developed for theevaluation of F[r, f] will be presented in Section VII.
C.

Comparison with the deterministic DFT equation(17)
dF½r; 0rlex ½r ¼ r
dr(18)

where DFdft[r, f] signifies the free-energy diﬀerence between
F[r, f] and F[r, 0], and the Lagrange multiplier lN enforces the8814 | Soft Matter, 2021, 17, 8810–8831(22)

It has been proved in various ways that the stochastic DFT
equation (eqn (14)) is converted into the deterministic DFT
equation44 when neglecting the additional free-energy functional DFdft[r, f]; hence, eqn (24) reduces towhich further readslex ½r; f ¼

(21)

givingÐlex ½r ¼ lex ½r; fÐ
Dflex ½r; feDFdft ½r;f
Ð;
DfeDFdft ½r;fdðr  r0 Þ cðr  r0 Þ;rðrÞ

Eqn (21) and (22) manifest that equilibrium DFT is incorporated into stochastic DFT.
It follows from eqn (16)–(18) thatlex ½r ¼

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
Zðr; tÞ
where z½r;~
Z can be expressed as z½r;~
Z ¼ r2Drðr; tÞ~using the bare diffusion constant D and the vectorial white
noise field ~Zðx; tÞ defined by the correlation hZl(r, t)Zm(r 0 , t 0 )i =0dlmd(r  r )d(t  t 0 ), and lex[r] will be referred to as the nonequilibrium excess chemical potential.
Combining eqn (10) and (15), we havedF½r; f F½r;f
D½r
Df
e
Ð dr
;
lex ½r ¼
DfeF½r;f D½r(20)w1(r  r 0 ) = r(r){d(r  r 0 ) + h(r  r 0 )r(r 0 )}.

The Fokker–Planck equation for P[r, t] is equivalent to thestochastic DFT equation, or the Dean–Kawasaki equation,44
which is given bydA½r
lex ½r ¼;
drd2 F½r; 0
¼ fðrÞfðr0 Þ ¼ wðr  r0 Þ;drðrÞdrðr0 Þ(13)Non-equilibrium excess chemical potential kex[q]@rðr; tÞZ;
¼ r Drrlex ½r þ z½r;~
@t

(19)

using the density–density correlation function w1(r  r 0 ) (seeAppendix A6 for details). For the concrete representation of theabove propagator w(r), we define the DCF c(r) and the total
correlation function h(r) based on equilibrium DFT. The propagator w(r) is expressed as

according to equilibrium DFT.58–61
B.

DFdft ½r; f ¼ F½r; f  F½r; 0ðð1
drdr0 fðrÞw1 ðr  r0 Þfðr0 Þ;
¼
2

(25)

when the last noise term on the rhs of eqn (14) disappears.
Going back to eqn (20) and (21), we find that the Ramakrishnan–Yussouff functional57 of the intrinsic Helmholtz freeenergy F[r, 0] is of the following form:

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F½r; 0 ¼ F½n; 0 ðð1
drdr0 DrðrÞcðr  r0 ÞDrðr0 Þ þ DFid ½r;2

ð
Fid ½r ¼ drrðrÞfln rðrÞ  1g;(26)

The diﬀerence between the equilibrium and stochastic DFTs
can be clearly seen from plugging eqn (30) into eqn (14) and(29). On the one hand, the deterministic DFT equation(eqn (29)) ensures that eqn (30) is a steady-state condition: wehave @rdft @t ¼ 0 because the rhs of eqn (29) vanishes due to
rdF½r ; 0=dr  ¼ jrmj ¼ 0. On the other hand, the stochasdft

where Dr  r  n, DFid[r]  Fid[r]  Fid[n], and n = N/V denotes
the uniform mean density with V being the system volume.
Combining eqn (13) and (26) provides
ðm ¼ ln rðrÞ þ cdft ðrÞ  dr0 cðr  r0 ÞDrðr0 Þ;(27)which reads
Ð 000rðrÞ ¼ emcdft ðrÞþ dr cðrr ÞDrðr Þ ;(28)stating that the conventional relation of equilibrium DFT for aprescribed density r(r) is satisfied by adjusting the externalpotential cdft(r) (see also eqn (A7)).
We obtain from plugging eqn (26) into eqn (25) the deterministic DFT equation using the Ramakrishnan–Yussouff functional:@rðr; tÞdF½r; 0
¼ r Drr@t
drð

¼ Dr2 rðr; tÞ  r Drðr; tÞ dr0 rcðr  r0 ÞDrðr0 ; tÞ:(29)Comparison between eqn (14) and (29), or between eqn (24) and(25), indicates the difference between the stochastic and deterministic DFT equations.
D.

Defining metastable states based on stochastic DFT

Before considering the metastability condition for the stochastic DFT equation (eqn (14)), we connect the metastable distribution rdft ðrÞ determined by equilibrium DFT with thedeterministic DFT equation (eqn (29)). In the absence of theexternal field (i.e., cdft  0), eqn (13) reduces to the metastability condition for equilibrium DFT:
dF½r; 0
¼ m;
(30)
drðrÞ r¼r
dft




where F rdft ; 0 becomes equal to the intrinsic Helmholtz freeenergy defined in equilibrium. Correspondingly, eqn (27) leads toðm ¼ ln rdft ðrÞ  dr0 cðr  r0 ÞDrdft ðr0 Þ:(31)
The non-equilibrium excess chemical potential lex[r] should
disappear at rdft : 
dF½r; 0
lex rdft ¼
lN
dr r¼r(32)dft¼ m  lN ¼ 0;
implying that the Lagrange multiplier lN is correctly identified withthe equilibrium chemical potential m at rdft , as mentioned above.

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dft

tic DFT equation (eqn (14)) for rdft ðr; tÞ becomes@rdft ðr; tÞ
dDFdft ½rdft ; f
þ z rdft ;~Z ;¼ r Drdft r
@t
drdft

(33)

revealing that, in general, rdft ðr; tÞ is not a steady-state distribution in terms of stochastic DFT.
Meanwhile, the metastability condition for the stochastic
DFT equation (eqn (14)) is that the metastable excess chemical
potential lex[r*] given by eqn (23) does not necessarily vanish
but has a spatially constant value lex :
dA½r
¼ lex :
(34)
dr r¼r
The first term on the rhs of eqn (14) disappears when eqn (34)
is satisfied, yielding@r ðr; tÞ@t¼ hz½r ;~Zi ¼ 0;(35)

on noise-averaging. A previous study based on stochastic thermodynamics has shown that the heat dissipated into thereservoir is negligible on average when satisfying eqn (34) or
eqn (35).54
In this study, we thus adopt the metastability condition (34)
based on stochastic DFT, instead of eqn (30).
E. The zero-wavevector structure factor S*(0) in a metastablestate defined by eqn (34)
It has been demonstrated near and at jamming that thestructure factor S*(k) in a metastable state can be written as
S ðkÞ ¼

1hr ðkÞihr ðkÞi;N(36)

because the structure factor in a frozen state mainly arises fromthe configurational part which is associated with the averaged
positions of arrested particles.42 Eqn (32), (34) and (36) imply
that S*(0) is obtained from the metastable chemical potential,l ¼ lex þ lN ;(37)

in a similar manner to equilibrium DFT as follows:1dl ¼ n 
S ð0Þ
dr r ¼nð
¼ 1  n drc ðrÞð¼  c ð0Þ  n(38)

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rs

(39)

ð
c ð0Þ  n

2

4pr drc ðrÞ;rs

where the metastable DCF c*(r) is defined by eqn (38) using themetastable chemical potential l* and the approximate expression given in the last line of eqn (39) is obtained from theOrnstein–Zernike equation regarding c*(r) at zero separation r =0 (see Appendix B1 for detailed derivation).
Equilibrium DFT, on the other hand, provides the metastable density distribution rdft ðrÞ determined by eqn (30). It
follows that the metastable structure factor S*(k) reads
S ðkÞ ¼

1
r ðkÞrdft ðkÞ;
N dft

(40)

with hr*(k)i in eqn (36) being replaced by rdft ðkÞ. Then, we
obtain from eqn (26) and (30)

1
dm 
¼n
S ð0Þ
drdft r ¼n
dft
(41)
ð
¼ 1  n drcðrÞ;metastable DCF (the second requirement (ii)) remains to beinvestigated.
In reality, the zero-separation DCF tends to have an extremely large value near freezing in repulsive sphere systems; forinstance, the Percus–Yevick approximation of hard sphere
fluids provides93

kB Tc ð0Þ ¼

Table 1 classifies the hyperuniform and non-hyperuniform
systems into four types for clarifying the non-hyperuniform
state to be addressed hereafter. The type H1 in Table 1 signifies
the hyperuniform state without requirement for c*(0) becauseeqn (4) is completely satisfied (i.e., Lc - N).
Despite the finiteness of the long-range nature, eqn (39) still
predicts that the hyperuniformity of type H2 is necessarily
observed near and at jamming unless the zero-separation
divergence of c*(0) is avoided. This is because 1/S*(0) diverges
due to either the long-range nature or the divergent behavior atzero separation, as found from combining eqn (4) and (39).
To summarize, there are two requirements on the nonhyperuniform DCF c*(r) of type N1 as follows:(i) Finiteness of the long-range nature – the nonhyperuniformity requires a drop in the long-ranged DCF forr 4 Lc. Namely, the first requirement is that c*(r) must decay
rapidly to zero for r 4 Lc;39–43 otherwise, the second term on
the rhs of eqn (39) is divergent.
(ii) Short-range cutoﬀ – as seen from the first term on the rhs
of eqn (39), the metastable DCF at zero separation (i.e., c*(0))
must have a finite value even as the densely packed systems
approach jamming, which is the second requirement.
Eqn (39) reveals that the zero-wavevector structure factor nevervanishes without meeting both of the above requirements. Nevertheless, exclusive attention in previous studies39–43 has been paid
to the former requirement, and the short-range cutoff of the8816 | Soft Matter, 2021, 17, 8810–8831(42)suggesting the divergent behavior of c*(0) in a frozen state.
Thus, we focus on the emergence of type N1 when investigating the degradation of hyperuniformity. To be more specific,
we show theoretically that the non-hyperuniformity of type N1satisfies(c ðrÞ C
1
r2

ðr ¼ 0Þðs o r  Lc Þ;(43)though the hyperuniformity of type H2 is incorporated into
equilibrium DFT as input:
81><cðrÞ  r1>:
r2

confirming that the DCF c(r) determines the zero-wavevector
structure factor.

IV. Our aim: non-hyperuniform stateson the target

@P
;
@n

ð0  r  sÞðs o r  Lc Þ;(44)where c(r) is diﬀerent from the completely hyperuniform DCF
in that Lc is supposed to have a finite value. Following equilibrium DFT, eqn (41) and (44) lead to S*(0) B c(0) 4 104
despite the finiteness of the long-range power-law decay, whichis the hyperuniformity of type H2.
We are now ready to address the issues on the nonhyperuniformity of type N1. In what follows, we present a
preliminary result obtained in the Gaussian approximation
for comparing stochastic and equilibrium DFTs, and subsequently prove in the strong-coupling approximation of stochastic DFT that eqn (44) transforms to eqn (43) as a result of theensemble average over the fluctuating f-field (see alsoeqn (17)).

V. Gaussian approximation ofstochastic DFT
In the first place, we investigate the free-energy functional
diﬀerence between the stochastic and equilibrium DFTs when
performing the Gaussian approximation given by eqn (19). In
the Gaussian approximation, eqn (23) reduces tol½r  lex ½r þ lN¼

ððdF½r; 0 1 dþ
drdr0 w1 ðr  r0 ÞfðrÞfðr0 Þ:
drðrÞ
2 dr

(45)

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As seen from Appendix A7 for details, we haveððddrdr0 w1 ðr  r0 ÞfðrÞfðr0 Þ
dr
Ð
d ÐÐ
drdr0 w1 ðr  r0 Þ DffðrÞfðr0 ÞeDFdft ½r;f
drÐ
¼
DfeDFdft ½r;f

(46)

Typical behaviors of the metastable DCF c*(r)As proved in the next section, the development of the strongcoupling expansion method, or the 1/g expansion method,
allows us to find the following form of the metastable chemical
potential l*:
ðwð0Þ
l ¼ ln r ðrÞ þ dr0 c ðr  r0 Þr ðr0 Þ;(51)2

cð0Þ  hð0Þ:
Combining eqn (26), (37) and (46), eqn (45) readsl¼ l½r  ¼ lex þ lN
ð1
¼ ln r ðrÞ  dr0 cðr  r0 ÞDr ðr0 Þ þ fcð0Þ  hð0Þg;
2

in a metastable state. We find from eqn (38) and (47)ð1
¼ 1  n drcðrÞ
S ð0Þ

0

(47)

(48)

Eqn (49) is, on the one hand, of the same form as the previous
results obtained from the Gaussian approximation in various
ways when l* = m.94,95 On the other hand, comparison between
eqn (28) with cdft  0 and eqn (49) indicates that eqn (49) is
identical to the self-consistent equation of r*(r) conventionallyused in equilibrium DFT when
(50)

when lN = m and lex = {c(0)  h(0)}/2. Eqn (50) reveals thatstochastic fluctuations create an additional contribution, thesecond term on the rhs of eqn (50), to the equilibrium chemical
potential m.

VI. Main results and comparison withsimulation results
To go beyond the Gaussian approximation, we need to explore
an expansion method adequate for strongly correlated spheresystems near and at jamming. One candidate is the virial-type
expansion that has proven to be applicable to inhomogeneousionic fluids at strong coupling.96 In the next section, we willverify that the virial-type expansion can apply also to theevaluation of lex[r] given by eqn (17), hence yielding themetastable DCF c*(r) other than c(r).
In this section, the obtained form of the metastable DCF
c*(r), which satisfies the relation (43), is presented in advance(Section VIA). Subsequently, the calculated value of S*(0) is
compared with the simulation results (i.e., eqn (2)) on the nonhyperuniform structure factor at jamming (Section VIB).

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(52)

Since the relation w(0) c 1 holds at jamming, eqn (52) leads toc*(0) = 1,

while neglecting dc(0)/dr, or the triplet DCF. Comparison
between eqn (41) and (48) confirms that no degradation of
hyperuniformity is induced by Gaussian potential fluctuations.
To see the correspondence with previous results, it is convenient to transform eqn (47) to
Ð 0
1
0
0
r ðrÞ ¼ el þ dr cðrr ÞDr ðr Þ2fcð0Þhð0Þg :(49)

1
l ¼ m þ fcð0Þ  hð0Þg
20

c*(r  r 0 ) = 1  ev(rr )w(rr ).

(53)

regardless of the repulsive potential form of v(r). Eqn (53)reveals that, in general, the metastable DCF c*(r) given by (52)satisfies the second requirement (or eqn (43)) for the nonhyperuniformity (see requirement (ii) in Section IV). Particularly for hard spheres, the resulting form (52) reads
c ðrÞ ¼

1
ð~r  1Þ
cðrÞ ð~
r
1Þ;

(54)

where r̃  r/s. Eqn (54) meets the above-mentioned nonhyperuniformity requirements given by eqn (43) with C ¼ 1.
While the short-range cutoﬀ is seen in the third term on therhs of eqn (51), the second term on the rhs of eqn (51)corresponds to the eﬀective self-energy which is divergent due
to the power-law behavior expressed by eqn (44). This implies
that the eﬀective self-energy term (= w(0)/2) oﬀsets the decreasein the interaction contribution due to the short-range cutoﬀ.
Thus, we have obtained various forms of the chemical
potential given by eqn (31), (47) and (51) from the equilibrium
DFT, the stochastic DFT in the Gaussian approximation, andthe stochastic DFT in the strong-coupling approximation,
respectively. The above discussions also suggest that the different results of the hyperuniform and non-hyperuniform
chemical potentials (i.e., eqn (31) and (51)) are compatible witheach other in terms of absolute values.
In Fig. 1, comparison is made between the r̃-dependencies ofc(r) and c*(r) for the repulsive harmonic potential given byv(r) = e(1  r̃)2Y(1  r̃),(55)

where e controls the interaction strength and Y(x) is the
Heaviside step function. It is supposed in Fig. 1 that c(r) isof the following form:8a>ð~r  1Þ
>
>
r~ 
>

>
<b
Lc
~
1
o
r

(56)
cðrÞ ¼ r~2
>

 s
>
>
>
L
c
>
:0
~
:
r 4
s
To be noted, eqn (56) does not include the delta function1 dð~r  1Þ due to the isostaticity, a significant negative con4f
tribution to c(r) at r̃ = 1.13

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Fig. 1 Comparison between the metastable DCF c*(r) given by eqn (52) and the hyperuniform DCF c(r) expressed by eqn (56) for the parameter sets of (e,a, b) as follows: while we need to fix two parameters, a and b, for representing the expression (56) of c(r), it is necessary to set not only a and b, but also theparameter e of the original interaction potential v(r) given by eqn (55) for showing the obtained form (52) of c*(r). (a) A log–log plot of c(r) and c*(r) which
are depicted using the parameter sets as follows: (e, a, b) = (106, 10, 4) and (106, 1, 1). (b) A linear plot for comparing c(r) and c*(r) with the parameter set of(e, a) = (106, 10, 4) in more detail. (c) A semi-log plot of c*(r) when e is decreased from 106 to either 103 or 1. The metastable DCF c*(r) is softened with thedecrease of e in eqn (55).

Previous simulation studies38–43 have indicated the parameter ranges of e Z 104, a B 101 and 0.1 r b r 101 close
to jamming. Correspondingly, we consider four sets of parameters in Fig. 1: (e, a, b) = (106, 10, 4), (106, 1, 1), (103, 1, 1)
and (1, 1, 1). In Fig. 1(a), the hyperuniform and metastable
DCFs, c(r) and c*(r), are depicted for two sets of parameters,(e, a, b) = (106, 10, 4) and (106, 1, 1), on a log–log plot. We cansee from Fig. 1(a) that the potential value of the metastable
DCF saturates to unity irrespective of the short-range behaviorof c(r), and that the short-range deviation of c*(r) from c(r)is larger with the increase of a and b. A magnified view for r Z s
is shown in Fig. 1(b), allowing us to make a comparison
between c(r) and c*(r) for (e, a, b) = (106, 10, 4) in moredetail. Fig. 1(b) shows that c*(r) converges to c(r) for r c seven when there is an obvious difference in the DCFs at r = 2s
between c(r = 2s) = b/4 and c*(r = s) = 1  eb/4 for b = 4.
Fig. 1(c) compares the profiles of c*(r) for e = 103 and 1 with aand b being the same value (a = b = 1) on a semi-log plot. This
indicates that the metastable DCF inside the sphere (i.e., c*(r)for r r s) is not changed until the interaction strength
represented by the parameter e is reduced considerably(for instance, e = 1 in Fig. 1(c)) far from the jamming valuesof e Z 104.
B. Comparison with simulation results given by eqn (2) and(7)
It follows from eqn (39), (52), (55) and (56) that the approximate
form of the zero-wavevector structure factor S*(0) is determined

8818 | Soft Matter, 2021, 17, 8810–8831

by both the volume fraction fv of packed spheres and the cutoﬀ
length Lc:
 
1
Lc
24fv b(57)
S ð0Þsfor Lc/s c 1; see Appendix B2 for detailed derivation. The first
choice to investigate the type-N1 non-hyperuniformity at jamming is to set Lc/s = 10 and fv = 0.65, according to the previous
simulation results38–43 of non-hyperuniform harmonic-core
sphere systems. Eqn (57) then becomes 1/S*(0) = 156b, implyingthat relation (2) applies to the metastable structure factor:
102 o

1o 103 ;
S ð0Þ

(58)

with b  O½10b  being in a reasonable range of 0 r b o 1.
For validation of the above evaluation, Fig. 2 provides thedependences of 1/S*(0) on Lc/s in the range of eqn (7) for b = 1,
4 and 10 with fv = 0.65 being used as before. As seen from Fig. 2,
a comparison between the precise result (see eqn (B9) inAppendix B2) and the approximate expression (57) shows thateqn (57) is an acceptable approximation. The precise results
depicted by solid lines in Fig. 2 further verify the relation (58)for 1 r b r 10 in the range of eqn (7) for Lc/s. Thus, we findthat the metastable DCF c*(r) given by eqn (52), one of the main
results in this study, quantitatively explains the previous simulation results on the non-hyperuniformity of type N1.
It is also suggested by Fig. 2 that b B 101 leads to 1/S*(0) o2
10 as long as the cutoﬀ of c*(0) = 1 holds. This result appears

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We introduce the rescaled propagator w̃(r) using a coupling
constant g:
~
wðrÞ
¼

wðrÞ
;
g2

(59)

wð0Þ
g¼e 2 :

Fig. 2 The last expression in eqn (39) can be calculated analytically whenusing eqn (52) and (56). The three solid lines depict the analytical result (B9)with eqn (B7) and (B8), or the precise results of the zero-wavevector
structure factor S*(0), for b = 1, 4 and 10 at fv = 0.65. For comparison, thedotted lines representing the approximate form (57) are also drawn for thesame parameter sets: b = 1, 4 and 10 at fv = 0.65. The yellow area
corresponds to the non-hyperuniform range of 1/S*(0) which is given byeither eqn (2) or eqn (58).

to contradict previous simulation results13 in hyperuniform
hard sphere systems where not only the small value ofb B 101 but also the existence of Lc in the range of eqn (7)have been found. At the same time, however, the divergent
relation c(r) E 10/r̃ (r̃ o 1) has been verified for the present
hyperuniform hard sphere systems.13 Accordingly, the divergent behavior of the hyperuniform DCF c(0) at zero-separation
ensures the hyperuniform relation (3): the relation, 1/S(0) E
c(0) 4 104, holds even when b B 101 and Lc/s B 101, whichis exactly the hyperuniform state of the type-H2 in Table 1.

Since we consider the type-H2 hyperuniform systems as mentioned before, it is found from eqn (21) and (44) that the
coupling constant g is approximated by g E ec(0)/2 andbecomes extremely large near and at jamming.
We aim to develop the 1/g expansion method at strong
coupling (g c 1), provided that g is extremely large but isfinite. In the next subsection, we will show that the virial-type
expansion method, the density-expansion method, can be
regarded as the 1/g expansion method. Before proceeding, we
see the g-dependencies of functionals based on the following
criteria:~ B g0,
Criterion 1: DFdft[~r, f]Ð~1 ðr0  r00 Þ ¼ dðr  r00 Þ.
~  r0 Þw
Criterion 2: dr0 wðr
Criterion 1 allows us to discern the perturbative terms atstrong coupling, in comparison with the rescaled functional~ B g0, whereas criterion 2 is equivalent to theDFdft[~r, f]
Ornstein–Zernike equation58–61 for rescaled correlation functions, c̃(r) and h̃(r), that should be defined to satisfy~  r0 Þ ¼
wðrdðr  r0 Þ c~ðr  r0 Þ;~ ðrÞ
rr(r 0 )},w̃1(r  r 0 ) = r~(r){d(r  r 0 ) + h̃(r  r 0 )~

(60)

(61)

consistently with the original definitions given by eqn (21) and(22).
It is found from eqn (19) and (59) that criterion 1 imposes
potential rescaling as follows:~f(r) = gf(r).

(62)

VII. Verification of the main result
given by eqn (51) and (52) in the strongcoupling approximation

On the other hand, it follows from criterion 2, or eqn (59) to(61), that the correlation functions and the density field arenecessarily rescaled as

There are three steps to verify both the metastable chemical
potential l* and the DCF c*(r) given by eqn (51) and (52),respectively. First, we define the coupling constant g andpresent the free-energy functionals rescaled by g, suggesting
the validity of the strong-coupling expansion method, or thedensity-expansion method at high density (Section VIIA). Second, the non-equilibrium chemical potential l[r] defined byeqn (16)–(18) is calculated for non-interacting spheres at strong
coupling (Section VIIB). Third, we connect the 1/g expansion,
which is equivalent to the density expansion (or the fugacity
expansion96), with the virial-type term expressed by the Mayer
function, thereby proving eqn (51) and (52) (Section VIIC).

~
hðrÞ ¼ g2 hðrÞ;This journal is © The Royal Society of Chemistry 2021cðrÞ ¼ g2 c~ðrÞ;
(63)

~ðrÞ
r
rðrÞ ¼ 2 ;g
to satisfy criterion 2.
Combining eqn (59) and (62) transforms eqn (19) to
h
i 1ðð
~ w
~ 0 Þ;~~1 ðr  r0 Þfðr~; f ¼
drdr0 fðrÞ(64)
DFdft r2
meeting criterion 1. Meanwhile, the rescaled form F[~r, 0] of the
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~; 0 ¼ F ½n~; 0 
F ½r

1
2g2

ðð

f 0 Þ þ DFid ½r
f cðr  r0 ÞDrðr
~;
drdr0 DrðrÞ~

ð
~ðrÞ
~ðrÞ
r
r
~ ¼ dr 2 ln 2  1 ;Fid ½r
g
g
(65)

f r
~  n~ and n  n~ g2 . Comparison between thewhere Dr
rescaled functionals, eqn (64) and (65), suggests that the~-dependent terms can be treated perturbatively at strongr
coupling (g c 1).
B. The non-equilibrium chemical potential of non-interacting
spheres at strong coupling
Going back to eqn (12), we develop the 1/g expansion method. It
is found from eqn (28) thatmþifðri Þcdft ðri Þewhich is the functional to be evaluated using the 1/g expansion
method at strong coupling, g c 1.
Let us see the non-equilibrium chemical potential l[r] in areference system of non-interacting spheres, prior to formulating the strong-coupling approximation of eqn (71). In theabsence of the interaction potential v(ri  rj), eqn (71) is exactly
reduced to the ideal free-energy functional for the noninteracting system:ð~ðrÞ ig~jðrÞr
Fnon ½r; j ¼ F0 ½r; j  dr
e
g
8
9
~
g2 wð0Þð<1 r~ ðrÞ wð0Þ~
~ ðrÞ e 2 þig~jðrÞ =
ij
þ

¼ dr~
rðrÞ 2 ln 2 þ
:g
;
g
2
g
g2
ð
~
g2 wð0Þ1~   2 dr~rðrÞe 2 þig~jðrÞ :r; j¼ F0 ½~
g(72)DcðrÞþifðrÞ

¼ rðrÞe
;ðDcðrÞ ¼  dr0 cðr  r0 ÞDrðr0 Þ:(66)

It follows that
~ 
r; j
lnon ½~

dFnon ½r; j
drAlso, we shift the fluctuating-potential field from f to jsuch that

¼ g2

wð0Þ
þ ijðrÞ ¼ DcðrÞ þ ifðrÞ;
2

¼ ln

(67)

whose rescaled form is~
g2 wð0Þf~þ ig~jðrÞ ¼ DcðrÞ
þ igfðrÞ;2(68)

due to the relations (62) and (63). Substituting eqn (68) intoeqn (66), we have the rescaled form,wð0Þemþifðri Þcdft ðri Þ ¼ rðrÞe 2 þijðrÞ ¼~ðrÞ ig~jðrÞ
r
e
;
g

(69)

wð0Þ
~ðrÞ=g. Moreover, eqn (28),considering that rðrÞe 2 ¼ grðrÞ ¼ r

ð
¼ drrðrÞfln rðrÞ þ DcðrÞ þ ifðrÞ  mgg

eig~jðri Þ~
g2 wð0ÞjðrÞ~
g2 wð0Þ
2 þig~
de
þig~
jðrÞ
2
~ðrÞ
:
r
e

d~
rðrÞ

~ over ther, j]We need to perform the average of lnon[~f-field based on the original definition in addition to relation (68). In the strong-coupling approximation, we obtainfrom eqn (73)~  ¼ ln rðrÞ þlnon ½r ¼ lnon ½~
r; j

wð0Þ
;
2

(74)

In the strong-coupling approximation, the long-range correlations of the shifted fluctuating potential j(r) are maintained:jðrÞjðr0 Þ ¼ wðr  r0 Þ;Yevðri rj Þ ;

i; j

(71)

8820 | Soft Matter, 2021, 17, 8810–8831

(73)

~ ðrÞ
~
g2 wð0Þ
r
þ ig~
jðrÞ
þ1þ
2
g
2

C. Connecting the 1/c expansion with the virial-type
expansion: derivation scheme of eqn (51) and (52)Combining eqn (66)–(70), eqn (12) readsÐÐeF ½r;f¼jþiDcþ drrðrÞm ¼ eF0 ½r;jþ drrðrÞm

i~g2 wð0ÞjðrÞ
2 þig~

dr~
rðrÞe

(70)

ð
wð0Þ
¼ drrðrÞ ln rðrÞ þþ ijðrÞ  m2
ð F0 ½r; j  drrðrÞm:

 Tr

ð

which corresponds to the non-equilibrium chemical potential
lnon[r] of non-interacting spheres; see Appendix C2 for thedetailed derivation of eqn (74). Eqn (74) implies that
ð
wð0Þ
:
(75)
Fnon ½r ¼ Fid ½r þ drrðrÞ
2(67) and (69) are arranged to giveð drrðrÞfcdft ðrÞ  ifðrÞg

Yr~ðri Þ

~
dF0 ½~
r; j
d

d~
r
d~
rðrÞ

(76)

as well as relation (20) for the fluctuating f-potential (see thederivation of eqn (C15) in Appendix C1). Eqn (76) implies that ~ðri Þr
~ ðri Þeig~jðri Þ  ¼ r
:
(77)

 g
g

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Hence, the 1/g expansion becomes equivalent to the following
density expansion (or the fugacity expnasion96):
Tr

Yr~ðri Þ
i¼

g

eig~jðri Þevðri rj Þ

l½r ¼ lnon ½r þ Dl½r

i; j

ð
1
X
1
N¼0

Y

N!

ð
dr1

drN

N
Y
~ðri Þ
ri¼1

time by definition; see Appendix C for detailed and more precise
discussions regarding the derivation of eqn (82). We obtain fromeqn (74) and (82)gig~jðri Þ

eY¼ ln rðrÞ þvðri rj Þ

e

ð n
(83)
o
wð0Þ
0
0
þ dr0 1  evðrr Þwðrr Þ rðr0 Þ:
2

i; j

ð
1
1þ
dr~
rðrÞeig~jðrÞgðð
1~ðr1 Þ~dr1 dr2 r
rðr2 Þevðr1 r2 Þþig~jðr1 Þþig~jðr2 Þ
þ
2g

The main result given by eqn (51) and (52) is thus verified.
(78)

VIII. Discussions

1
~ ;
r; j
 1 þ U½~
g
where
ð
Ð
~  ¼ dr~U½~
r; jrðrÞeig ds~jðsÞ^r1 ðsÞ
(79)
ðð
Ð
1
vðr1 r2 Þþig ds~
jðsÞ^
r2 ðsÞ
~ ðr1 Þ~dr1 dr2 rþrðr2 Þe
:2g
In eqn (79), we have introduced instantaneous one- and two2P
^2 ðsÞ ¼
dðr  ri Þ, forparticle densities, r^1(s) = d(s  r) and r

In this section, we aim to gain insight into the short-range cutoﬀ of
the metastable DCF c*(r) from dynamic aspects. We consider afluctuating displacement field u(r, t) which is related to the density
diﬀerence, n(r, t) = r(r, t)  r*(r). Since the fluctuating density fieldn(r, t) obeys the linearized Dean–Kawasaki equation of stochastic
DFT,44,50–52,54–56 the short-range dynamics of u(r, t) can be inferredfrom the n-field dynamics. First, we will see that the short-range
cutoff of the metastable DCF implies the disappearance of theinteraction-induced restoring force against the fluctuating density
field n(r, t) (Section VIIIA). Next, the connection of the short-range
softening with anharmonic soft modes will be discussed in termsof u-field dynamics (Section VIIIB). Last, we summarize the resultspresented so far using Table 2 (Section VIIIC).

i¼1

making a distinction between the first and the second terms on
the rhs of eqn (79).
Combining eqn (71), (72) and (78) provides
1~  ¼ F0 ½~~ ;~   ln 1 þ U½~
F½~
r; j
r; j
(80)
r; j
g
and we define the non-equilibrium chemical potential diﬀerenceDl[r] due to the addition of the interaction potential v(r) as follows:
Dl½r ¼ g2

~
dF½~
r; j
 lnon ½r:d~rðrÞ

A. Dynamic implication for the short-range cutoﬀ of themetastable DCF c*(r)Expanding the non-equilibrium excess chemical potential lex[r]around r*(r), the Dean–Kawasaki eqn (14) becomes@rðr; tÞ@fnðr; tÞ þ r ðrÞg
¼
@t
@t
(
)

ð
dlex ½r
0¼ r Drr lex þ dr0nðr;tÞþ z½r;~
Z
drðr0 Þ r¼r

(81)

ð

¼ r Drr
It follows from eqn (72), (74), (76), (79) and (80) that the strongcoupling approximation of eqn (81) leads to
Dl½r ¼ 1 (84)

dðr  r0 Þ c ðr  r0 Þ;
¼
r ðrÞ

¼ 1

(82)

ð
Ð
0
~ðr0 Þeig ds~jðsÞfr^1 ðsÞþ^r1 ðsÞg þ O½g1 þ dr0 r
ð n
o
 
0
0
¼ dr0 1  evðrr Þwðrr Þ rðr0 Þ þ O g2 ;where it is noted that the instantaneous one-particle densities, r^1(s)= d(s  r) and rb0 1 ðsÞ ¼ dðs  r0 Þ, are unable to coexist at the same

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nðr0 ; tÞ þ z½r;~
Z;
r¼r



dlex ½r
dðr  r0 Þ
dlN 
0

c
¼
ðr

r
Þ

drðr0 Þr¼r
drðr0 Þ r¼r
r ðrÞ

~ =d~
gdU½~
r; j
rðrÞ
~ =g
1 þ U½~r; j~  dU½~
~
gdU½~
r; j
r; j
~  þ O½g1 
þ
U½~
r; j
d~
rðrÞ
d~
rðrÞ
Ð
¼ 1  geig ds~jðsÞ^r1 ðsÞðÐ~ðr2 Þevðr1 r2 Þ eig ds~jðsÞ^r2 ðsÞ dr2 rdlex ½r
dr0
drðr0 Þ 

(85)



where rlex  ¼ 0 has been used in eqn (84). Combining Eqn (84)with eqn (85) leads to the linearized Dean–Kawasaki equation as
follows:@nðr; tÞ
¼ Dr2 nðr; tÞ
@t
(86)
ð
pﬃﬃﬃ
Z;
 r Dr dr0 rc ðr  r0 Þnðr0 ; tÞ þ 2z½r ;~due to the manipulation of the noise term.54 Eqn (86) represents
the overdamped dynamics of the fluctuating density field n(r, t)around the metastable non-hyperuniform state.

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Eqn (86) indicates that the interaction-induced restoring
force against the density deviation n(r, t) is given by the sumof rc*(r  r 0 )n(r 0 , t). Focusing on the short-range contribution
to this force, we find that microscopic environments in thehyperuniform and non-hyperuniform states are quite diﬀerent
from each other. While the scaling behavior (6) in a hyperuniform state predicts the divergence of |rc(r  r 0 )| - N in thelimit of |r  r 0 | - 0, the short-range cutoﬀ of the nonhyperuniform DCF c*(r) creates the opposite situation on theparticle scale:|rc*(r  r 0 )| E 0 (|r  r 0 | o s),(87)

as seen from Fig. 1. The above relation implies that there is nointeraction-induced restoring force against the n-field in the
non-hyperuniform states at the particle-scale while preserving
the long-range contribution, rc*(r  r 0 )n(r 0 , t) = r(b/|r 
r 0 |2)n(r 0 , t), for |r  r 0 | c s.

B.

Then, let u(r, t) be the fluctuating displacement field
induced by the fluctuating density field n(r, t). In the first
approximation, we have97
n(r, t) = r {r*(r)u(r, t)}.

(88)

Combining eqn (54), (56) and (86)–(88), we can verify that thedisplacement field u(r, t) shares common features with that of
the quasi-localized soft modes as follows:
The interaction-induced restoring force of u(r, t) is longranged in correspondence with recent simulations98–101
because of the power-law decay of the metastable DCF c*(r)represented by eqn (54) and (56).
Eqn (86)–(88) imply the short-range anharmonicity of u(r,t) at the particle scale.
This connection of our theoretical results (particularly,eqn (87)) with the quasi-localized soft modes suggests thatthe virial-type expansion in a glassy state represents the particle–particle interactions occurring due to the intermittent
particle rearrangements.

Microscopic mechanism behind the appearance of eqn (87)

Eqn (86) for the overdamped Brownian dynamics is insuﬃcientfor the descriptor of vibrational modes due to the absence of
the inertia term, and yet eqn (87) suggests the emergence ofdynamic softening in non-hyperuniform systems. We can learn
the microscopic mechanism of soft modes from previous
studies on quasicontacts of a contact network, a skeleton of
jammed matter.18–22,75,76 The previous findings could provide
an intuitive understanding of the virial-type expansion at highdensity as will be seen below.
For isostatic and hyperuniform systems, the packing geometry uniquely defines the contact forces as well as the spatial
network structures including void distributions.1,2,8–12 Previous
studies have shown that the isostatic and hyperuniform
states disappear upon relaxing the strict constraints on thesize- and spatial-distributions of voids slightly away fromjamming.1,2,9,10,12 The relative abundance of non-isostatic contacts provides quasicontacts that carry weak forces, thereby
creating local excitations with little restoring forces.12,19–22,76
The possible particles forming the quasicontacts include
rattlers and/or bucklers.12,19–22,76–83 It has been found, forinstance, that the bucklers in the d-dimensional space, havingd + 1 contacts as part of the contact network, are likely to bebuckled to generate quasi-localized soft modes observed in thelowest-frequency regime.19–22,76–83 The anomalous vibrationalmodes have been shown to exhibit strong anharmonicities thatare accompanied by intermittent rearrangements of particles asfollows: opening a weak contact of a buckler (i.e., buckling)yields a disordered core of a few particle scale with a power-law
decay of displacements which are coupled to the elastic background of the contact network.19–37,76–83
It is not the center of our concern whether or not the rattlers
and/or bucklers significantly contribute to the quasicontacts
to degrade the hyperuniformity. It is, however, illuminating
to interpret eqn (86) and (87) in terms of quasi-localized
soft modes.

8822 | Soft Matter, 2021, 17, 8810–8831

C. Summarizing the results in comparison with othertreatments
The diﬀerences in the free-energy density functionals between
the equilibrium and stochastic DFTs are summarized asfollows:(i) The density functional A½r appearing in the metastabilityeqn (34) represents the free-energy functional of the given
density distribution r(r), instead of the equilibrium freeenergy functional F[r, 0]. It is a clear advantage over equilibrium DFT that stochastic DFT can make use of the fieldtheoretic formulation in obtaining A½r.
(ii) The metastability eqn (34) states that the functional
derivative dA½r=dr should yield a spatially constant lex , which
has been referred to as the metastable excess chemical
potential. The sum of lex and the Lagrange multiplierlN corresponds to the metastable chemical potential andis reduced to the equilibrium chemical potential (i.e.,lex þ lN ¼ m) when lex ¼ 0 and lN = m in equilibrium; see also
the discussion after eqn (32).
(iii) As found from eqn (30), (41) and (56), the input of thehyperuniform DCF allows equilibrium DFT to predict thehyperuniformity of a metastable state, without the knowledge
on the reference density distribution in an amorphous state.
These characteristics of stochastic DFT enable us to evaluate
the extent to which fluctuations around the metastable density
r* aﬀect the metastable chemical potential l*. Actually, we havedemonstrated that stochastic DFT is relevant to determine
metastable states around a hyperuniform state. Stochastic
DFT provides the analytical form of the metastable DCF thathas a short-range cutoﬀ inside the sphere while retaining thelong-range power-law behavior. We should keep in mind thatthe long-range hyperuniform behavior is preserved because theGaussian weight, eDFdft, for the virial-type expansion premisesthat non-hyperuniform states considered are located near a
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Fig. 3 A schematic comparison of theoretical approaches presented inthis study. The spatially uniform density is identically n as shown on thevertical axis, and the hyperuniformity is incorporated into equilibrium DFT
by inputting the hyperuniform DCF c(r) given by eqn (56); nevertheless, wehave hyperuniform and non-hyperuniform treatments shown in blue andorange, respectively. The diﬀerent results are due to the distinct values of
non-interacting reference free-energy functionals (i.e., Fid[r] given byeqn (26) and Fnon[r] given by eqn (75)), which is represented by thehorizontal axis.

DCF yields the zero-wavevector structure factor in quantitative
agreement with previous simulation results1,2,38–43 of degradedhyperuniformity.
Moreover, both Fig. 3 and Table 2 summarize the results bycomparing the following theoretical approaches discussed so
far: the equilibrium DFT using the Ramakrishnan–Yussouff
free-energy functional,57 the stochastic DFT in the Gaussian
approximation (see Section V), and the stochastic DFT in thestrong-coupling approximation (see Section VII).
The vertical axis in Fig. 3 shows that the density distribution
considered has the same density n on average. The diﬀerence isattributed to the inhomogeneous distributions around n: thehyperuniform density distribution rdft ðrÞ, shown in blue, satisfies eqn (3) for the inverse of the zero-wavevector structure
factor 1/S*(0), whereas the non-hyperuniform range of density
distribution r*(r), shown in orange, satisfies eqn (2). As summarized in Fig. 3, the equilibrium DFT and stochastic DFT in
the Gaussian approximation provide hyperunifomity, whereas
the stochastic DFT in the strong-coupling approximation provides non-hyperuniformity.
Meanwhile, the transverse axis of Fig. 3 shows that the above
two types of theoretical approaches take distinct reference freeenergy functionals, as found by comparing Fid and Fnon given byeqn (26) and (75), respectively. In the hyperuniform theories, on
the one hand, the ideal free-energy functional Fid/N per particle
is the reference functional for the evaluation of interactionenergy (see eqn (26)), following the conventional treatment ofequilibrium DFT,57–61 where N denotes the total number ofspheres as before. On the other hand, as a reference functionalof stochastic DFT in the strong-coupling approximation, weThis journal is © The Royal Society of Chemistry 2021

Soft Matter
used the free-energy functional Fnon/N of a non-interacting
system per particle that is larger than the ideal one Fnon/N by
the self-energy w(0)/2. It can be stated that a perturbation field
theory method becomes more relevant to the evaluation ofintermittent fluctuations due to the increase in the reference
free energy.
Table 2 presents the more detailed classification of thehyperuniform and non-hyperuniform theories. There are two
types of classifications for the above three treatments. One
classification is based on the DFT type of whether the dynamical DFT relies on the deterministic equation (eqn (29)) or thestochastic equation (eqn (14)). The former approach represented by eqn (29) is equivalent to equilibrium DFT as clarified
at the beginning of Section IIID, whereas the latter eqn (14)forms the basis of the last two stochastic approaches where theadditional contribution DF[r, f] to the intrinsic Helmholtz freeenergy F[r, 0] is to be considered. The other aspects of theoretical classification concern the predictability of nonhyperuniformity especially when the hyperuniformity is incorporated into the DCF (i.e., eqn (44) or (56)) of equilibrium DFT
as input. The type specification column in Table 2 indicatesthat the stochastic DFT in the Gaussian approximation falls
into the same category (type H2 defined in Table 1) of theequilibrium DFT in this light.
As confirmed from Table 2, the use of the density-expansion
method at strong coupling is indispensable to convert thehyperuniform structure factor at the zero wavevector into thenon-hyperuniform one satisfying the simulation results given
by eqn (2). The outstanding feature of the non-hyperuniform
DCF c*(r) is the short-range cutoﬀ, thereby predicting theabsence of the interaction-induced restoring force for the
short-range dynamics (i.e., eqn (87)).

IX. Concluding remarks
For comparison purposes, let us go back to the previous study15
where the degradation of perfect hyperuniformity in crystals,
quasicrystals, and disordered packings has been demonstrated
both theoretically and numerically, using the three scenarios ofimperfections (see Section II for the list of scenarios). The
second scenario (ii), which has been our concern, attributes
the violation of hyperuniformity to the stochastic occurrence of
spatially correlated displacements.
Combining eqn (76) and the dynamical discussion in Section VIII suggests that the above-mentioned second scenario forthe degradation of hyperuniformity is similar to the underlying
physics described by the averaged virial-type interaction term,
the third term on the rhs of eqn (51), under the long-rangecorrelated j-field; for it seems plausible that the long-rangecorrelated potential field arises from the elastic nature of thecontact network. From the discussion, we infer that the virialtype degradation of hyperuniformity reflects the intermittent
rearrangements of particles and is more likely to be found in
the collectively jammed packings allowing for the shear deformations as mentioned before, rather than in the strictly

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jammed ones.1,2,8 In other words, type H2 defined in Table 1
corresponds to the collectively jammed packings, whereas type
H1 corresponds to the strictly jammed packings.
It remains to be seen whether the present formulation canbe extended to address the non-hyperuniform behaviors ofother measures than the density–density structure factor,
which are obtained from various physical quantities including
the local number variance for a window17 and the contact
number fluctuations.102,103 We also envision that advancing
stochastic DFT44–56 at strong coupling will pave the way for therealistic description of quasi-localized soft modes induced by
the intermittent rearrangements of particles such as
bucklers.19–22,76–83

Conflicts of interest

We evaluate the underlined term in eqn (A3) with the help of
the Fourier transform of the delta functional as follows:

Tr

Y
i

em

Y

evðri rj Þ

Y^N ðrÞ  rðrÞ
d½r

r

i; j

ð
Ð
Y
Y
emþicðri Þevðri rj Þ e dricðrÞrðrÞ
¼ DcTr

(A4)

i; j

ð
Ð
¼ DceO½ic dricðrÞrðrÞ :Appendix A: details on the constrained
free-energy functional A½r
1. Verification of eqn (9)The distribution functional P[r, t] defined by eqn (8) satisfies
the Fokker–Planck equation as follows:44,104
ð@P½r; tdd dA½r¼  dr r Drrþ
P½r; t;
(A1)
@t
dr
dr
dr
from which we find that eqn (9) satisfies the stationary condition qPst[r]/qt = 0. It has also been shown that eqn (A1) isequivalent to the Dean–Kawasaki eqn (14).44,104
2. Definition of A½rIn eqn (A1) as well as in eqn (14), the canonical ensemble isnaturally required for the free-energy functional A½r of a given
density r because we consider the overdamped dynamics of thedensely packed sphere system with the total number N ofspheres being fixed. Hence, A½r is defined using the configurational integral for the canonical ensemble as follows:ððY1eA½r ¼
dr1
drN
evðri rj Þ
N!
i; j
(A2)
Y
^ N ðr; tÞ  rðr; tÞ:d½rr

Yet, equilibrium DFT, a key ingredient in this study, needs to be
formulated in the grand canonical system. We therefore write
A½r with the help of the contour integral over a complex
variable z = em:54,104I
1dzeA½r ¼
2pi zNþ1
0
1
Y
Y Y
m
vðr
r
Þ
^N ðr; tÞ  rðr; tÞA;
 @Tr
e
e i j
d½r
i; j

3. Derivation of eqn (13)iThere are no conflicts to declare.

iso that the canonical ensemble may be recovered after
performing the grand canonical ensemble represented by
1 1 Ð
Ð
P
Tr 
dr1
drN .
N¼0 N!

r(A3)

8824 | Soft Matter, 2021, 17, 8810–8831

The c-field is separated into the fluctuating potential field f(r)and the saddle-point field icdft(r):c(r) = f(r) + icdft(r),(A5)

where cdft(r) is determined by the saddle-point equation,dðO½icÞ¼ irðrÞ:dcðrÞ c¼icdft(A6)

The functional diﬀerentiation on the left-hand side of eqn (A6)
provides the density in equilibrium of the system under theexternal field cdft. Denoting the equilibrium density byh^rN(r)ieq, the saddle-point eqn (A6) implies thath^rN(r)ieq = r(r).

(A7)

The above relation states that a prescribed density r(r) is
equated with the equilibrium density due to the potential cdft
along the saddle-point field.
For later convenience, we also introduce the intrinsic Helmholtz free energy F[r, 0], the central functional of equilibrium
DFT:58–61ðð
F½r; 0  drrðrÞm  O½cdft   drrðrÞcdft ðrÞ;(A8)

showing that the intrinsic Helmholtz free energy F[r, 0]
is defined by the first Legendre transform of the grandpotential O[cdft] with the saddle-point field cdft(r) being
applied. Therefore, the identity (13) is satisfied as well as thatin equilibrium DFT.

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4. Derivation of eqn (12)Combining eqn (A4) and (A5), we haveY
Y Y
^ N ðrÞ  rðrÞemevðri rj Þ
d½r
Tr
i

where the basic relationship has been used in equilibrium DFT
as follows:

r

i; j

ð
Y
Y
emþifðri Þcdft ðri Þevðri rj Þ¼ DfTr
i

ð

i; j

Ð
 e drrðrÞfcdft ðrÞifðrÞg
ðÐ
¼ DfeO½cdft ifþ drrðrÞfcdft ðrÞifðrÞg
ð

(A9)

dr0



d2 F½r; 0
d2 O
¼ dðr  r00 Þ;0000drðrÞdrðr Þ dcðr Þdcðr Þc¼icdft(A16)

as well as the definition (20) of w(rr’).

Ð¼ DfeF½r;fþ drrðrÞm ;7. Derivation of eqn (46)when definingðF½r; f  drrðrÞm  O½cdft  ifð drrðrÞfcdft ðrÞ  ifðrÞg

It is found from eqn (22) that eqn (46) reads(A10)

as an extension of eqn (A8). Eqn (A9) and (A10) validate
eqn (12).

ððddrdr0 w1 ðr  r0 ÞfðrÞfr0 Þ
dr
ð h

0
¼ dr0 w1 ðr  r0 Þ fðrÞfðr0 Þ

5. Derivation of eqn (10) and (11)It follows from eqn (A3) and (A9) that
I
ð
Ð
1
dz
DfeF½r;fþ drrðrÞm
eA½r ¼
Nþ12pi z
F½r;f

¼ Dfe

1
2pi

I

1
dz Ð
 drrðr;tÞþNþ1
z

(A17)

#
0
DfffðrÞfðr0 Þg eDFdft ½r;fÐþ w ðr  r Þ;
DfeDFdft ½r;f
1

ð

(A15)

0

Ð

where
!

 1
0
w ðr  r0 Þ

(A11)

ð

dðr  r0 Þ þ 2hðr  r0 Þrðr0 Þ¼¼ DfeF½r;f D½r;
where

(A18)

neglecting the density dependence of the total correlation
function h(r  r 0 ), and eqn (A17) and (A18) give a precise
definition of {f(r)f(r 0 )} 0 appearing in eqn (A17).
In eqn (A17), we use the following approximation:

I

1
1
dz Ð
D½r 
 drrðr;tÞþNþ1
2pi
8
ð z

>
>
>1
drrðrÞ ¼ N
>
<
¼
ð

>
>
>
>
0
drrðrÞaN
::w1 ðr  r0 Þþ hðr  r0 Þrðr0 Þ;
rðrÞ

(A12)

Eqn (A11) and (A12) verify eqn (10) and (11), respectively.

nfðrÞfðr0 Þ

o0

d

drðrÞ
Ð
¼

Ð

DffðrÞfðr0 ÞeDFdft ½r;f
Ð
DfeDFdft ½r;f0DfffðrÞfðr0 Þg eDFdft ½r;f
Ð
DfeDFdft ½r;f

6. Derivation of eqn (19)The diﬀerence DFdft[r, f] = F[r, f]  F[r, 0] between eqn (A8)and (A10) reads
ð(A13)
DFdft ½r; f ¼ O½cdft  if  O½cdft  þ drirðrÞfðrÞ:The quadratic expansion of O[cdft  if] around the imaginary
saddle-point field icdft yields

ðð

1
d2 O

fðrÞfðr0 Þ(A14)
DFdft ½r; f ¼
drdr0
2
dcðrÞdcðr0 Þc¼cdft

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dDFdft ½r; f
drðrÞ
(
)
dDFdft ½r; fþ fðrÞfðr0 ÞdrðrÞ fðrÞfðr0 Þ

Ð

(A19)

0

DfffðrÞfðr0 Þg eDFdft ½r;f
Ð
:
DfeDFdft ½r;f

Accordingly, eqn (A17) reduces toSoft Matter, 2021, 17, 8810–8831 | 8825

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d
dr

ðð

Paper

drdr0 w1 ðr  r0 ÞfðrÞfðr0 Þ

2. Derivation of eqn (57)ð

It is noted that c*(r)h*(r) decays far more rapidly than c*(r) evenwhen the eﬀective correlation functions converge to the hyperuniform ones, c(r) and h(r), for r 4 s. Therefore, eqn (39) becomesð Lc1
 c ð0Þ  n 4pr2 drc ðrÞ
S ð0Þ
s
ð
6fv Lc(B4)¼  c ð0Þ  3 4pr2 drc ðrÞ
ps sð Lc =sd~rr~2 c ð~rÞ;¼  c ð0Þ  24fvh
0
dr w1 ðr  r0 Þ fðrÞfðr0 Þ
0

n
o0 
þ w1 ðr  r0 Þ fðrÞfðr0 Þ

(A20)

ð
1
w1 ð0Þþ hð0Þ  dr0 hðr  r0 Þrðr0 Þcðr0  rÞ  2
rðrÞ
r ðrÞ
ð
¼  dr0 hðr  r0 Þrðr0 Þcðr0  rÞ ¼ cð0Þ  hð0Þ;
¼

1

where

n
o0
0
fðrÞfðr0 Þ ¼ fwðr  r0 Þg¼

3

dðr  r0 Þ cðr  r0 Þ
rðrÞ


0

(A21)

dðr  r0 Þ;
r2 ðrÞ

neglecting the density dependence of the DCF, and the Ornstein–
Zernike equation has been used in the last equality of eqn (A20).

Appendix B: the zero-wavevector
structure factor S(0) represented by
the metastable DCF c*(r)1. Derivation of eqn (39)
We consider the Ornstein–Zernike equation for the metastablecorrelation functions, c*(r) and h*(r), in a uniform state. The
Ornstein–Zernike equation at zero separation is expressed asðh ð0Þ ¼ c ð0Þ þ n 4pr2 drc ðrÞh ðrÞð¼ c ð0Þ  n

4pr2 drc ðrÞ

(B1)

rs

ð
þn

4pr2 drc ðrÞh ðrÞ;rs

where the relationship h*(r) = 1 (0 r r r s) has been used inÐ
the above second line. Adding the term n rs 4pr2 drc ðrÞ onboth sides of eqn (B1), we haveð
4pr2 drc ðrÞh ð0Þ  nwhich will be used in calculating the last term in the last line ofeqn (B4). Integration by parts yields
ð Lc =sð3
Lc =s Lc =s 3 dc ð~
rÞ
3
d~
r~
r2 c ð~rÞ ¼  ~r c ð~
rÞ 1 þ
d~
rr~
d~
r
1
1
 3  ð Lc =sb
Lc
Lc
þ c ð1Þ þ 2bcd~rer~2
¼ 
s
s
1




Lc
Lc
þ 2bI2 b;;¼ I1 b;
s
s
(B6)
where


 3  
Lc
Lc
Lc
¼
þ c ð1Þ;
c
I1 b;
s
s
s

¼

ð1

4pr drc ðrÞh ðrÞ;rs

which readsð h ð0Þ  n drc ðrÞð
4pr drc ðrÞ  n2rs

4pr2 drc ðrÞh ðrÞ:
rs

(B3)

8826 | Soft Matter, 2021, 17, 8810–8831

ebx
x

2

#1
s=Lc

ð1
2b

2

dxebx

(B8)

s=Lc

2

¼

2

2

ebx
dx 2
x
s=Lc
"

(B2)

ð
ð
¼ c ð0Þ  n drc ðrÞ þ n(B7)

and the change of variable from r̃ to x = 1/r̃ givesð Lc =sbLc¼
d~re~r2
I2 b;
s
1

¼ 

rs

ð
¼ c ð0Þ  n

where the relationship fv = pns /6 between the volume fraction fv andthe spatially averaged density n has been used in the second line of theabove equation. For r̃ = r/s 4 1, combining eqn (52) and (56) provides


b
c ð~
(B5)
rÞ ¼ 1  exp  2 ;r~

Lc ebðs=Lc Þ ebspﬃﬃﬃ rﬃﬃﬃ
pﬃﬃﬃ
bs
p
:
b
erf
b  erf
Lcb

Combining eqn (B2)–(B8), we have




1
Lc
Lc
¼c ð0Þþ 8fv I1 b;
þ 16bfv I2 b;
:
s
s
S ð0Þ

(B9)

The approximate form of eqn (B9) for Lc/s c 1 leads to eqn (57):This journal is © The Royal Society of Chemistry 2021

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1
S ð0Þ



3

bðs=Lc Þ2

Lc
Lc ec ðLc =sÞ þ 16bfv
s
s
 
Lc
;
24fv b
s
 8fv

(B10)

where we have used the approximations, c*(Lc/s) E b(s/Lc)2
2
and Lceb(s/Lc) /s E Lc/s, in the above second line. Fig. 2 shows
the Lc/s-dependencies of 1/S*(0) in order to compare eqn (B9)and (B10).

Appendix C: details on the averaging
operation over the u-field in thestrong-coupling approximationÐ
where Sð0Þ ¼ 1 þ dr0 hðr  r0 Þrðr0 Þ ¼ 0 for the hyperuniform
structure factor S(0) at zero wavevector, and it is supposed thatw(0) is the position-independent function because of theneglect of the triplet DCF (see also the statement after eqn (48)).
Before proceeding to the f-field averaging operation definedby eqn (17), we clarify the corresponding terms where the strongcoupling approximation needs to be developed. To this end, let~ be the general functional given by the sum of the j-indepen~z[j]~~dent part zc and the remaining j-dependent
contribution zr[j]:~~ = zc + zr[j].
z[j]It is noted that we can validate the following expansion atstrong coupling (i.e., g c 1):i E ½~r;~j

11~  þ E 0 2 ½~
~iE 0 ½~
r; j
r; j
g
2g

1. Shifting the fluctuating-potential field from / to u ineqn (17): a general formulation and validation of eqn (76)e g0

We first see extra terms when DFdft[r, f] is represented by the
j-field. Eqn (67) is rearranged to give

1
~ ;
r; j
 1  E½~gðð
f 0Þ
~1 ðr  r0 Þ~~   drdr0 w
E 02 ½~
r; jjðrÞDcðr

wð0ÞfðrÞ ¼ jðrÞ þ iDcðrÞ  i
;
2

(C1)

which further reads

1


2~

g wð0Þ
~f;fðrÞ ¼ gfðrÞ
¼ g~jðrÞ þ iDcðrÞ
i
2

(C3)

Plugging eqn (C2) into eqn (19), we havehi1
~ ¼ DFdft ½~
f þ i E 0 ½~
~   2 DFdft ½~~r; jr; Dc
r; jDFdft r; f ¼ gf
g
g
ðð
h
i
f ¼ 1 drdr0 w
f Dcðrf 0 Þ;~; Dc~1 ðr  r0 ÞDcðrÞ
DFdft r2ðð
f 0 Þ:~  ¼ dsds0 w~1 ðs  s0 Þ~
r; jjðsÞDcðs
E 0 ½~(C4)ig~; g2 wð0Þ=2¼ 0 for the hyperuniform correlabecause of DFdft rhtion function h(r):
"
#

ð
ð
g
g2 wð0Þ
w2 ð0Þ
~;¼
DFdft rdrrðrÞ dr0 fdðr  r0 Þ þ hðr  r0 Þrðr0 Þg
2
8

ð
ð
w2 ð0Þ
drrðrÞ 1 þ dr0 hðr  r0 Þrðr0 Þ
8¼

Nw2 ð0ÞSð0Þ ¼ 0;
8Combining eqn (C4) and (C7) provides the approximate functional z½~
j that is averaged over the f-field, instead of the~
j-field,
based on the definition of eqn (17) as follows:
Dfz½~
jeDFdft ½r;f
Ð
DfeDFdft ½r;fÐ
jeDFdft ½r;f
D~
jz ½~¼ zc þ Ð r DF ½r;f
D~
je dft

z½~
j ¼

Ð

DF

1
~;~
½r
jþ DF~ i E ½~½~
r;Dc
r;~j

g 0
g2 dftje dftD~
jzr ½~¼ zc þ
1i~Ð~;~DF ½rr;DcgE 0 ½~
r;~jjþ 2 DFdft ½~
g
D~
je dftÐ1~  eDFdft ½r~;~jD~
jzr ½~
j 1  E½~
r; j
g


¼ zc þÐ1~  eDFdft ½r~;~jD~
j 1  E½~
r; j
g


Ð
1
~  eDFdft ½r~;~jD~
jzr ½~
j 1  E½~
r; j
g


¼ zc þ
8Ð
9
1
>
> De
>
~  eDFdft ½r~;~j >r; j
j 1  E½~
<
=
Ð
g
Ð
D~
jeDFdft ½r~;~j>>D~jeDFdft ½r~;~j
>
>
:
;



1
~  eDFdft ½r~;~jD~
jzr ½~
j 1  E½~
r; j
g

 :
¼ zc þ
Ð
1
~
DF
½
r
;~
j
~
D~
je dft  1  E½~
r; j
g
Ð

(C8)
(C5)

This journal is © The Royal Society of Chemistry 2021f 0 Þ:~1 ðs  s0 Þ~jðsÞDcðs
dsds0 wÐf
¼ DcðrÞ:

¼

ðð

(C7)

(C2)

due to the rescaling given by eqn (63) and its associated form,
ð
f 0Þ
Drðr
DcðrÞ ¼  dr0 g2 c~ðr  r0 Þ 2gð
f 0Þ
¼  dr0 c~ðr  r0 ÞDrðr

(C6)~  appearing in the lastHere, another averaging operation E½~
r; jSoft Matter, 2021, 17, 8810–8831 | 8827

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line of eqn (C8) has been introduced for representingÐ~ eDFdft ½r~;~j
D~
jE½~
r; j
Ð
~ 
E½~
r; j
;
D~
jeDFdft ½r~;~j(C9)which becomes equivalent to eqn (17) when replacing the ffield with the j-field; however, the notation of the above
average has been altered for revealing that there is a diﬀerence~  and E½~~  as shown by the general form (C8).
r; j
r; j
between E½~
We obtain from eqn (C7) and (C9)
1
1
1
~ ¼
~  þ E 02 ½~~
iE 0 ½~
E½~
r; j
r; j
r; j
g
g
2g
ðððð
1
~ ðrÞ~
drdr0 dsds0 j
¼ 2
jðsÞ
2g

In the remaining subsections, we will evaluate eqn (74) and(82) based on the above strong-coupling approximation represented by eqn (C14).
2. Derivation of eqn (74) from eqn (73)~ given by
Following the general form (C6), we classify lnon[~
r, j]
~
~
eqn (73) into the j-independent
part zc and the j-dependent~
contribution gz1[j]:~  ¼ zc þ gz1 ½~
lnon ½~r; jj;~
g2 wð0ÞjðrÞ
2 þig~~
g2 wð0ÞjðrÞ
2 þig~eÐ~
g2 wð0Þ^1 ðsÞþmðsÞg
jðsÞfr2 þig ds~;

guexp ½~j ¼ e~ ðrÞ ¼ 0;
j(C11)
~ ðrÞ~~  sÞ:jjðsÞ ¼ wðr
The strong-coupling approximation of the last equality ineqn (C8) further validates that
1
~   E½~
~
E½~
r; j
r; j
g
(C12)

which is available to calculate the third term on the rhs ofeqn (C14).
First, the Gaussian integration over the j-field yields~
g2 wð0ÞjðrÞ
2 þig~~
g2 wð0Þ¼ e~ ðrÞ~
drdr0 dsds0 fjjðsÞ  wðr  sÞg

2

þig~
jðrÞ

~ðrÞ
r

(C13)
f 0 Þw
f 0 Þ:~ ðr  r ÞDcðr
~1 ðs  s0 ÞDcðs
w
1

0

i
~ ðrÞ~~ ðrÞ~~ ðrÞ~~  þ O½g2 jjðr0 Þ ¼ j
jðr0 Þ  jjðr0 ÞE 0 ½~
r; jg
~  r0 Þ þ O½g2 ;¼ wðr
thereby justifying eqn (76).

8828 | Soft Matter, 2021, 17, 8810–8831

d~
rðrÞ

>
>
;

(C19)

~
g2 wð0Þg2 Ð Ð
0 Þ^~
dsds0 wðss
r1 ðsÞ^r1 ðs0 Þ
2 2

¼ e

¼  1;(C14)

because of g = eg2w̃(0)/2,

(C15)

>
>
:

Ð
~
g2 wð0Þig ds~
jðsÞ^
r1 ðsÞ
2 ez½~
j ¼ zc þ zr ½~j

according to the 1/g expansion method. Incidentally, it follows
from eqn (C14) that

8
9
2~
>
>
>
>
<deg wð0Þ
=
þig~
j
ðrÞ
2

¼ e

~  isr; j
Eqn (C12) implies that the above contribution DE 1 ½~
ignored. Combining eqn (C8) and (C12), we havei~  þ O½g2 ;¼ zc þ zr ½~
j  zr ½~
jE 0 ½~
r; j
g

0

~ðrÞe
gz1 ½~
j ¼  r

where
ðððð

(C17)

where a test density field m(s) is added to one-particle densityr^1(s) = d(s  r) that represents a single sphere located at r. The~ with respect to m(r) oﬀers
functional diﬀerentiation of uexp[j]the benefit of the expression (C17):~
g2 wð0Þ
gduexp ½~
j
¼ ig~
jðsÞe 2 þig~jðrÞ ;(C18)
dmðsÞ m¼0i
~  þ O½g2 ;
r; j¼ 1  E 0 ½~
g~ 
DE 1 ½~
r; j;:because ofi1~   2 DE 1 ½~
~
r; j
r; j
¼ 1  E 0 ½~g2g

0

where the definition of { } 0 is the same as that of eqn (A17).
For later convenience, we also introduce an extended form of

1~¼ 2 DFdft ½~
r; Dc
g

1

(C16)

~ðrÞe
j ¼ ig~jðrÞ  r
gz1 ½~(C10)

f 0 Þwf 0Þ
~1 ðs  s0 ÞDcðs
~1 ðr  r0 ÞDcðr
w

1
~
r; j
1  E½~g
1~1  E½~
r; j
g

~ðrÞ
~
g2 wð0Þ
r
þ1þ
;
g2
2

zc ¼ lnÐÐ~  s0 Þ^~
dsds0 wðsr1 ðsÞ^r1 ðs0 Þ ¼ wð0Þ,~
g2 wð0ÞjðrÞ
2 þig~¼ 1, and the similar approximation to eqn (A19).

e

Next, we investigate the third term on the rhs of eqn (C14),~ . The expression (C4) of E 0 ½~
~  implies thejE 0 ½~
r; jr; jor z1 ½~
necessity of evaluating the following contribution:2~
g wð0Þ
ii~ ðsÞgz1 ½~~ ðsÞe 2 þig~jðrÞ :jjðsÞ~jðrÞ  jj ¼ ~g
g(C20)

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It follows from eqn (C11) and (C18) that eqn (C20) reads!i
1 duexp ½~j ~ ðsÞgz1 ½~~  rÞ jj ¼ wðs
(C21)
 :
g
g dmðsÞ 

eqn (C26):~
Dl½r ¼ Dl½~
r; j

m¼0

~  þ O½g1 ¼ 1 þ gz1 ½~j þ z2 ½~j  iz1 ½~jE 0 ½~
r; jSince we havej þ O½g1 ¼ 1  1 þ z2 ½~~
g2 wð0Þg2 Ð Ð
0~0 ÞfmðsÞþ^r1 ðsÞgfmðs0 Þþ^
r1 ðs0 Þg
guexp ½~j ¼ e 2  2 dsds wðss

(C22)

ð
Ð
~ðr2 Þevðr1 r2 Þ eig ds~jðsÞ^r2 ðsÞ dr2 rin the presence of the test field m(r), the relation (C18) readsð
gduexp ½~
j^1 ðs0 Þgguexp ½~~  s0 Þfmðs0 Þ þ r¼  g2 ds0 wðs
j


dmðsÞ
m¼0

Ð
ig ds~
jðsÞ^r2 ðsÞ
de~ðr2 Þevðr1 r2 Þ r~ðr1 Þ
 dr2 rd~
rðr1 Þ
ð

m¼0~  rÞe
¼  g2 wðs

(C29)

~
g2 wð0Þg2 Ð Ð
0 Þ^~
dsds0 wðss
r1 ðsÞ^r1 ðs0 Þ
2 2

ð

0

0

n

Ð

b0 1 ðsÞ
^1 ðsÞþr
ig ds~
jðsÞ ro~ðr Þeþ dr r;~  rÞ:¼  g2 wðs(C23)This ensures that the first term on the rhs of eqn (C21) iscanceled by the second term:~ ðsÞgz1 ½~
j ¼ 0;j(C24)

thereby implying that~  ¼ 0:z1 ½~
jE 0 ½~
r; j

(C25)

Thus, we find from eqn (C14), (C16), (C19) and (C25)

where

we

results,gz1 ½~
j ¼ 1anddensity dependence of ew(0) = 1/g2 as before. It follows fromthe Gaussian integration when performing the averages in thelast equality of eqn (C29) that

g2

ÐÐ

0Þ
~
dsds0 wðssdðsr1 Þdðs0 r2 Þþ¼e¼ zc þ gz1 ½~j þ O½g1 

2Pdðsri Þdðs0 ri Þ
i¼1

2

2

2

~ 1 r2 Þg wð0Þ~
¼ eg wðr

(C26)

~ðrÞ
~
r
g2 wð0Þ
1
þ1þ
2
g
2

¼

~ðrÞ g2 wð0Þ
~
r
:
þ
g2
2

1 g2 wðre ~ 1 r2 Þ ;
g2
(C30)

Ð
deig ds~jðsÞ^r2 ðsÞ dðr1  r2 Þ g2 wðr¼
e ~ 1 r2 Þ ;~2 ðr1 Þrd~rðr1 Þ

The above last form is equivalent to the target expression (74).
3. Derivation of eqn (82)

(C31)

and

Eqn (81) reads~  ¼ 1 þ ig~
Dl½~
r; j
jðrÞ Ð~ =d~
gdU½~
r; j
rðrÞ
~ =g
1 þ U½~
r; j

(C27)

n

Ð

b0
~ðr0 Þeig ds~jðsÞr 1 ðsÞ r~ðrÞeig ds~jðsÞ^r1 ðsÞþ dr0 rn

g2 Ð Ð
0Þ r
b0 1 ðsÞrb0 1 ðs0 Þ^1 ðsÞ^~2
dsds0 wðss
r1 ðs0 Þþr

o

¼e

~
~
g2 wð0Þg2 wð0Þ
2  2

o0

¼
(C28)
;

where the definition of { } 0 is the same as that of eqn (A17).
The non-equilibrium chemical potential diﬀerence Dl[r] is~  over the j-field~obtained from averaging Dl½~
r; jin a similarmanner to the strong-coupling approximation adopted inThis journal is © The Royal Society of Chemistry 2021

e

o

¼ e

~ has been given in eqn (C16) andwhere z1[j]ðÐno0~ðr2 Þevðr1 r2 Þ r~ðr1 Þeig ds~jðsÞ^r2 ðsÞj ¼  dr2 r
z2 ½~
Ð

n

b0 1 ðsÞ
^ 1 ðsÞþr
ig ds~
jðsÞ rj þ z2 ½~j þ O½g1 ;¼ 1 þ gz1 ½~

ðthe~  ¼ 0, given by eqn (C19) and (C25), respectively,z1 ½~jE 0 ½~
r; j
8
n
o 9,
>
< igÐ ds~j r^ þrb0 >=1
1
and d edr ¼ 0 due to the neglect of the>>:;~  þ O½g1 
j  iz1 ½~jE 0 ½~
r; j
¼ zc þ gz1 ½~

¼ ln

used

Ð
g2 Ð Ð
0~0 Þ^
r2 ðsÞ^r2 ðs0 Þ
eig ds~jðsÞ^r2 ðsÞ ¼ e 2 dsds wðss~  ¼ zc þ gz1 ½~
lnon ½~
r; j
j;

ln

have

1
;
g2
(C32)

respectively. In eqn (C32), it is noted that the cross terms
r1 ðs0 Þ ¼ 0 because the one-particle
vanish: b
r1 ðsÞrb0 1 ðs0 Þ ¼ rb0 1 ðsÞbdensities, r^1(s) = d(sr) and rb0 1 ðsÞ ¼ dðs  r0 Þ, represent the
instantaneous densities of the target particle located at diﬀerent positions of r and r 0 and a single sphere is unable tosimultaneously exist at diﬀerent locations.

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Substituting eqn (C30)–(C32) into eqn (C29), we obtainð1~ðr2 Þevðr1 r2 Þwðr1 r2 Þ
Dl½r ¼  2 dr2 rgð
12~~ðr0 Þ
 evð0Þg wð0Þ
þ 2 dr0 r
g
(C33)
ð n
o
1
evð0Þ0
2~0Þ
~ ðr0 Þ  2¼ 2 dr0 1  evðrr Þg wðrr
r
g
g
ð n
o
 
0
0
¼ dr0 1  evðrr Þwðrr Þ rðr0 Þ þ O g2 ;hence verifying eqn (82).

Acknowledgements
The author thanks the anonymous referees for their valuablecomments and suggestions.

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