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Hiroshi
Frusawa

Laboratory of Statistical Physics, Kochi University of Technology, Tosa-Yamada, Kochi 782-8502, Japan. E-mail: frusawa.hiroshi@kochi-tech.ac.jp

Received
16th July 2021
, Accepted 5th September 2021

First published on 23rd September 2021

The disordered and hyperuniform structures of densely packed spheres near and at jamming are characterized 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 that exhibit 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 of stochastic density functional theory is relevant to explore the degradation mechanism. The strong-coupling expansion method of stochastic density functional theory is developed to obtain the metastable 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 cutoff inside the sphere while retaining the long-range power-law behavior. It is confirmed that the metastable DCF provides the zero-wavevector limit of the structure factor in quantitative agreement with the previous simulation results of degraded hyperuniformity. We can also predict the emergence of soft modes localized at the particle scale by plugging this metastable DCF into the linearized Dean–Kawasaki equation, a stochastic density functional equation.

Here we focus on computer glasses among the disordered hyperuniform systems. Recent methodological developments allow 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 emergence of hyperuniformity depends on the preparation protocols, partly because of the significantly long computational time that 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: the structure factor S(k) in a hyperuniform state exhibits a non-trivial linear dependence on the wavevector magnitude k in the low-wavevector range near and at jamming (i.e., S(k) ∼ k (k ≥ 0)), 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 the system scale.

Meanwhile, other simulation studies near and at jamming^{38–43} provide the non-vanishing structure factor at the zero wavevector. The degradation of hyperuniformity is that either saturation or an upturn is observed for S(k) at the lowest values of k, despite the linear relation above the crossover wavevector k_{c}:^{38–43}

(1) |

The quantitative difference between the non-hyperuniform and hyperuniform states can be seen from the inverse of the zero-wavevector structure factors. While the non-vanishing values of S(0) due to the incomplete linear-dependence of S(k) are in the range of^{38–43}

(2) |

(3) |

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 long-ranged 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:^{1,2,5–7,13,17}

(4) |

(5) |

(6) |

It follows from eqn (1) and (5) that the non-hyperuniform DCF c(r) at jamming satisfies the scaling relation (4) over a finite range. In other words, the violation of hyperuniformity occurs while maintaining the long-range behavior to a length scale L_{c}: eqn (4) holds in the range of σ < r ≤ L_{c}^{38–43} where

(7) |

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 emergence and 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 of eqn (7). A key ingredient of our formulation is the strong-coupling 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 packings near and at jamming.

The remainder of this paper consists of two parts. In the former part of Sections II–IV, the problem to be addressed is defined. Section II provides the theoretical background as to why stochastic DFT should be brought into the problem of fluctuation-induced non-hyperuniformity. In Section III, the basic 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 λ* 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.

System | Characterization of the DCF | Type | |
---|---|---|---|

Power-law decay | Magnitude at zero separation | ||

Hyperuniform | Complete | — | H1 |

Incomplete | Divergent | H2 | |

Non-hyperuniform | Incomplete | Finite | N1 |

Absent | Finite | N2 |

We see from the system specification that the non-hyperuniformity 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 > L_{c}. 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 and discussion regarding the metastable DCF c*(r). Before entering the main results, Section V compares the stochastic DFT with the equilibrium DFT^{57–74} in terms of S*(0). It is demonstrated as a preliminary result that the resulting forms of S*(0) in the equilibrium and stochastic DFTs coincide with each other as far as the Gaussian approximation of stochastic DFT is adopted. Section VI provides the metastable DCF expressed by the Mayer-type function form, hence verifying the cutoff for the metastable DCF c*(r) inside the sphere. As a consequence, we confirm that S*(0) satisfies relation (2), instead of eqn (3). In Section VII, the coupling constant γ to represent the strength of interactions is introduced using the hyperuniform DCF c(r), and it is shown that the 1/γ expansion method becomes equivalent to the virial-type one at the strong coupling of γ ≫ 1. Correspondingly, the interaction term in the metastable chemical potential λ* 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 the particle scale: the interaction-induced restoring force against density fluctuations around a metastable state vanishes within the scale of spherical diameter σ. 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 for comparing equilibrium DFT, stochastic DFT in the Gaussian approximation, and stochastic DFT in the strong-coupling approximation. The final remarks are given in Section IX.

Theory | State description | Statics | Dynamics | ||||
---|---|---|---|---|---|---|---|

DFT type | Approximation | Type | μ or λ* | DCF | Equation | Short-range | Long-range |

Equilibrium | Ramakrishnan–Yussouff | H2 | Eqn (31) | Eqn (44) or (56) | Eqn (29) | Frozen | Correlated |

Stochastic | Gaussian | H2 | Eqn (47) | Eqn (44) or (56) | Eqn (14) or (86) | Frozen | Correlated |

Stochastic | Strong-coupling | N1 | Eqn (51) | Eqn (52) or (54) | Eqn (14) or (86) | Soft | Correlated |

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 of weighing jammed states.^{1,2,5–17} Recently, however, the protocol-dependency problem is theoretically tackled: the canonical ensemble 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 types of jamming are hierarchical in that local and collective 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 the packing 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 the isostatic 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 mechanical rigidity of jammed packings is a necessary but not sufficient condition for hyperuniformity.

It has been conjectured that any strictly jammed saturated infinite packing of identical spheres is hyperuniform; the conjecture excludes the existence of rattlers or particles that are 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 other than the strictly jammed ones, including the isostatic and collectively 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 mechanical marginal stability.^{1,2,12,18–37,75–83}

Recent simulation results have demonstrated that thermal fluctuations in the marginal states are accompanied by the intermittent rearrangements of particles.^{18–37} As a consequence, the marginal systems become responsive to have low-frequency 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 of particles undergoing displacements.^{18–37} The low-frequency soft modes exhibit similar behaviors, and the common features of 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 the ensemble of configurations visited by the slow dynamics could reveal the characteristics of marginal stability associated with the free-energy landscape, even though possible configurations depend on a protocol adopted.^{1,2,5–14,16}

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 the appearance 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 ρ*(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 the metastable state of a liquid having an inhomogeneous and aperiodic density as a local minimum of the equilibrium free-energy functional with respect to the variation in the width parameter for the 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 for demonstrating 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 the second 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 expected that the above-mentioned second scenario (i.e., (ii) stochastic and spatially correlated displacements) could be described in terms 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 powerful tools for describing slowly fluctuating and/or intermittent phenomena, such as glassy dynamics, nucleation or pattern formation of colloidal particles, dielectric relaxation of Brownian dipoles, and even tumor growth (see ref. 44 for a thorough review). The stochastic density functional equation, which has often 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 great practical use is the Dean–Kawasaki equation linearized with respect 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 the metastable states considering fluctuations as clarified below. In Section VIII, stochastic DFT will also shed light on the dynamical properties of non-hyperuniformity.

First, the constrained free-energy functional is represented by the hybrid form using the functional and configurational integrals (Section IIIA). Next, we introduce the non-equilibrium 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 λ*, thereby yielding S*(0) expressed by c*(r) (Section IIID).

(8) |

As detailed in Appendix A, P[ρ, t] satisfies the Fokker–Planck equation given by eqn (A1). It follows from the stationary condition ∂P_{st}[ρ]/∂t = 0 on the Fokker–Planck equation that the distribution functional in a steady state, P_{st}[ρ], is determined by the free-energy functional of a given density field ρ(r, t):

(9) |

(10) |

(11) |

(12) |

As clearly seen from Appendix A, the fluctuating potential field ϕ(r) in eqn (12) traces back to the auxiliary field for the Fourier transform of the Dirac delta functional .^{54,55,92} Also, the functional F[ρ, ϕ ≡ 0] in the absence of the ϕ-field corresponds to the intrinsic Helmholtz free energy in the presence of the external field ψ_{dft}(r). Therefore, the following relation holds:

(13) |

(14) |

(15) |

Combining eqn (10) and (15), we have

(16) |

(17) |

(18) |

In the Gaussian approximation of the ϕ-field, we have

(19) |

(20) |

(21) |

w^{−1}(r − r′) = ρ(r){δ(r − r′) + h(r − r′)ρ(r′)}. | (22) |

It follows from eqn (16)–(18) that

(23) |

(24) |

(25) |

Going back to eqn (20) and (21), we find that the Ramakrishnan–Yussouff functional^{57} of the intrinsic Helmholtz free energy F[ρ, 0] is of the following form:

(26) |

(27) |

(28) |

We obtain from plugging eqn (26) into eqn (25) the deterministic DFT equation using the Ramakrishnan–Yussouff functional:

(29) |

(30) |

(31) |

(32) |

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 because the rhs of eqn (29) vanishes due to . On the other hand, the stochastic DFT equation (eqn (14)) for becomes

(33) |

Meanwhile, the metastability condition for the stochastic DFT equation (eqn (14)) is that the metastable excess chemical potential λ_{ex}[ρ*] given by eqn (23) does not necessarily vanish but has a spatially constant value :

(34) |

(35) |

In this study, we thus adopt the metastability condition (34) based on stochastic DFT, instead of eqn (30).

(36) |

(37) |

(38) |

(39) |

Equilibrium DFT, on the other hand, provides the metastable density distribution determined by eqn (30). It follows that the metastable structure factor S*(k) reads

(40) |

(41) |

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 at zero separation, as found from combining eqn (4) and (39).

To summarize, there are two requirements on the non-hyperuniform DCF c*(r) of type N1 as follows:

(i) Finiteness of the long-range nature – the non-hyperuniformity requires a drop in the long-ranged DCF for r > L_{c}. Namely, the first requirement is that c*(r) must decay rapidly to zero for r > L_{c};^{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 never vanishes without meeting both of the above requirements. Nevertheless, exclusive attention in previous studies^{39–43} has been paid to the former requirement, and the short-range cutoff of the metastable DCF (the second requirement (ii)) remains to be investigated.

In reality, the zero-separation DCF tends to have an extremely large value near freezing in repulsive sphere systems; for instance, the Percus–Yevick approximation of hard sphere fluids provides^{93}

(42) |

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 N1 satisfies

(43) |

(44) |

We are now ready to address the issues on the non-hyperuniformity 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 the ensemble average over the fluctuating ϕ-field (see also eqn (17)).

(45) |

(46) |

(47) |

(48) |

To see the correspondence with previous results, it is convenient to transform eqn (47) to

(49) |

(50) |

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 non-hyperuniform structure factor at jamming (Section VIB).

(51) |

−c*(r − r′) = 1 − e^{−v(r−r′)−w(r−r′)}. | (52) |

−c*(0) = 1, | (53) |

(54) |

While the short-range cutoff is seen in the third term on the rhs 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 decrease in 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, and the 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 with each other in terms of absolute values.

In Fig. 1, comparison is made between the -dependencies of −c(r) and −c*(r) for the repulsive harmonic potential given by

v(r) = ε(1 − )^{2}Θ(1 − ), | (55) |

(56) |

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 (ε, α, β) as follows: while we need to fix two parameters, α and β, for representing the expression (56) of c(r), it is necessary to set not only α and β, but also the parameter ε 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: (ε, α, β) = (10^{6}, 10, 4) and (10^{6}, 1, 1). (b) A linear plot for comparing c(r) and c*(r) with the parameter set of (ε, α) = (10^{6}, 10, 4) in more detail. (c) A semi-log plot of c*(r) when ε is decreased from 10^{6} to either 10^{3} or 1. The metastable DCF c*(r) is softened with the decrease of ε in eqn (55). |

Previous simulation studies^{38–43} have indicated the parameter ranges of ε ≥ 10^{4}, α ∼ 10^{1} and 0.1 ≤ β ≤ 10^{1} close to jamming. Correspondingly, we consider four sets of parameters in Fig. 1: (ε, α, β) = (10^{6}, 10, 4), (10^{6}, 1, 1), (10^{3}, 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, (ε, α, β) = (10^{6}, 10, 4) and (10^{6}, 1, 1), on a log–log plot. We can see from Fig. 1(a) that the potential value of the metastable DCF saturates to unity irrespective of the short-range behavior of −c(r), and that the short-range deviation of −c*(r) from −c(r) is larger with the increase of α and β. A magnified view for r ≥ σ is shown in Fig. 1(b), allowing us to make a comparison between −c(r) and −c*(r) for (ε, α, β) = (10^{6}, 10, 4) in more detail. Fig. 1(b) shows that −c*(r) converges to −c(r) for r ≫ σ even when there is an obvious difference in the DCFs at r = 2σ between −c(r = 2σ) = β/4 and −c*(r = σ) = 1 − e^{−β/4} for β = 4. Fig. 1(c) compares the profiles of −c*(r) for ε = 10^{3} and 1 with α and β being the same value (α = β = 1) on a semi-log plot. This indicates that the metastable DCF inside the sphere (i.e., −c*(r) for r ≤ σ) is not changed until the interaction strength represented by the parameter ε is reduced considerably (for instance, ε = 1 in Fig. 1(c)) far from the jamming values of ε ≥ 10^{4}.

(57) |

(58) |

For validation of the above evaluation, Fig. 2 provides the dependences of 1/S*(0) on L_{c}/σ in the range of eqn (7) for β = 1, 4 and 10 with f_{v} = 0.65 being used as before. As seen from Fig. 2, a comparison between the precise result (see eqn (B9) in Appendix B2) and the approximate expression (57) shows that eqn (57) is an acceptable approximation. The precise results depicted by solid lines in Fig. 2 further verify the relation (58) for 1 ≤ β ≤ 10 in the range of eqn (7) for L_{c}/σ. Thus, we find that 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.

Fig. 2 The last expression in eqn (39) can be calculated analytically when using 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 β = 1, 4 and 10 at f_{v} = 0.65. For comparison, the dotted lines representing the approximate form (57) are also drawn for the same parameter sets: β = 1, 4 and 10 at f_{v} = 0.65. The yellow area corresponds to the non-hyperuniform range of 1/S*(0) which is given by either eqn (2) or eqn (58). |

It is also suggested by Fig. 2 that β ∼ 10^{−1} leads to 1/S*(0) < 10^{2} as long as the cutoff of −c*(0) = 1 holds. This result appears to contradict previous simulation results^{13} in hyperuniform hard sphere systems where not only the small value of β ∼ 10^{−1} but also the existence of L_{c} in the range of eqn (7) have been found. At the same time, however, the divergent relation −c(r) ≈ 10/ ( < 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) ≈ −c(0) > 10^{4}, holds even when β ∼ 10^{−1} and L_{c}/σ ∼ 10^{1}, which is exactly the hyperuniform state of the type-H2 in Table 1.

(59) |

We aim to develop the 1/γ expansion method at strong coupling (γ ≫ 1), provided that γ is extremely large but is finite. In the next subsection, we will show that the virial-type expansion method, the density-expansion method, can be regarded as the 1/γ expansion method. Before proceeding, we see the γ-dependencies of functionals based on the following criteria:

Criterion 1: ΔF_{dft}[, ] ∼ γ^{0},

Criterion 1 allows us to discern the perturbative terms at strong coupling, in comparison with the rescaled functional ΔF_{dft}[, ] ∼ γ^{0}, whereas criterion 2 is equivalent to the Ornstein–Zernike equation^{58–61} for rescaled correlation functions, (r) and (r), that should be defined to satisfy

(60) |

^{−1}(r − r′) = (r){δ(r − r′) + (r − r′)(r′)}, | (61) |

It is found from eqn (19) and (59) that criterion 1 imposes potential rescaling as follows:

ϕ(r) = γ(r). | (62) |

(63) |

Combining eqn (59) and (62) transforms eqn (19) to

(64) |

(65) |

(66) |

(67) |

(68) |

(69) |

(70) |

(71) |

Let us see the non-equilibrium chemical potential λ[ρ] in a reference system of non-interacting spheres, prior to formulating the strong-coupling approximation of eqn (71). In the absence of the interaction potential v(r_{i} − r_{j}), eqn (71) is exactly reduced to the ideal free-energy functional for the non-interacting system:

(72) |

(73) |

(74) |

(75) |

(76) |

(77) |

(78) |

(79) |

Combining eqn (71), (72) and (78) provides

(80) |

(81) |

(82) |

(83) |

(84) |

(85) |

(86) |

Eqn (86) indicates that the interaction-induced restoring force against the density deviation ν(r, t) is given by the sum of −∇c*(r − r′)ν(r′, t). Focusing on the short-range contribution to this force, we find that microscopic environments in the hyperuniform and non-hyperuniform states are quite different from each other. While the scaling behavior (6) in a hyperuniform state predicts the divergence of |∇c(r − r′)| → ∞ in the limit of |r − r′| → 0, the short-range cutoff of the non-hyperuniform DCF c*(r) creates the opposite situation on the particle scale:

|∇c*(r − r′)| ≈ 0 (|r − r′| < σ), | (87) |

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 the size- and spatial-distributions of voids slightly away from jamming.^{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, for instance, that the bucklers in the d-dimensional space, having d + 1 contacts as part of the contact network, are likely to be buckled to generate quasi-localized soft modes observed in the lowest-frequency regime.^{19–22,76–83} The anomalous vibrational modes have been shown to exhibit strong anharmonicities that are accompanied by intermittent rearrangements of particles as follows: 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.

Then, let u(r, t) be the fluctuating displacement field induced by the fluctuating density field ν(r, t). In the first approximation, we have^{97}

ν(r, t) = −∇·{ρ*(r)u(r, t)}. | (88) |

• The interaction-induced restoring force of u(r, t) is long-ranged in correspondence with recent simulations^{98–101} 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 that the virial-type expansion in a glassy state represents the particle–particle interactions occurring due to the intermittent particle rearrangements.

(i) The density functional appearing in the metastability eqn (34) represents the free-energy functional of the given density distribution ρ(r), instead of the equilibrium free-energy functional F[ρ, 0]. It is a clear advantage over equilibrium DFT that stochastic DFT can make use of the field-theoretic formulation in obtaining .

(ii) The metastability eqn (34) states that the functional derivative should yield a spatially constant , which has been referred to as the metastable excess chemical potential. The sum of and the Lagrange multiplier λ_{N} corresponds to the metastable chemical potential and is reduced to the equilibrium chemical potential (i.e., ) when and λ_{N} = μ in equilibrium; see also the discussion after eqn (32).

(iii) As found from eqn (30), (41) and (56), the input of the hyperuniform DCF allows equilibrium DFT to predict the hyperuniformity 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 ρ* affect the metastable chemical potential λ*. Actually, we have demonstrated that stochastic DFT is relevant to determine metastable states around a hyperuniform state. Stochastic DFT provides the analytical form of the metastable DCF that has a short-range cutoff inside the sphere while retaining the long-range power-law behavior. We should keep in mind that the long-range hyperuniform behavior is preserved because the Gaussian weight, e^{−ΔFdft}, for the virial-type expansion premises that non-hyperuniform states considered are located near a hyperuniform state. As confirmed in Section VI, the obtained DCF yields the zero-wavevector structure factor in quantitative agreement with previous simulation results^{1,2,38–43} of degraded hyperuniformity.

Moreover, both Fig. 3 and Table 2 summarize the results by comparing 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 the strong-coupling approximation (see Section VII).

Fig. 3 A schematic comparison of theoretical approaches presented in this study. The spatially uniform density is identically n as shown on the vertical axis, and the hyperuniformity is incorporated into equilibrium DFT by inputting the hyperuniform DCF c(r) given by eqn (56); nevertheless, we have hyperuniform and non-hyperuniform treatments shown in blue and orange, respectively. The different results are due to the distinct values of non-interacting reference free-energy functionals (i.e., F_{id}[ρ] given by eqn (26) and F_{non}[ρ] given by eqn (75)), which is represented by the horizontal axis. |

The vertical axis in Fig. 3 shows that the density distribution considered has the same density n on average. The difference is attributed to the inhomogeneous distributions around n: the hyperuniform density distribution , 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 ρ*(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 free-energy functionals, as found by comparing F_{id} and F_{non} given by eqn (26) and (75), respectively. In the hyperuniform theories, on the one hand, the ideal free-energy functional F_{id}/N per particle is the reference functional for the evaluation of interaction energy (see eqn (26)), following the conventional treatment of equilibrium DFT,^{57–61} where N denotes the total number of spheres as before. On the other hand, as a reference functional of stochastic DFT in the strong-coupling approximation, we used the free-energy functional F_{non}/N of a non-interacting system per particle that is larger than the ideal one F_{non}/N by the self-energy w(0)/2. It can be stated that a perturbation field theory method becomes more relevant to the evaluation of intermittent fluctuations due to the increase in the reference free energy.

Table 2 presents the more detailed classification of the hyperuniform 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 the stochastic 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 the additional contribution ΔF[ρ, ϕ] to the intrinsic Helmholtz free energy F[ρ, 0] is to be considered. The other aspects of theoretical classification concern the predictability of non-hyperuniformity 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 indicates that the stochastic DFT in the Gaussian approximation falls into the same category (type H2 defined in Table 1) of the equilibrium DFT in this light.

As confirmed from Table 2, the use of the density-expansion method at strong coupling is indispensable to convert the hyperuniform structure factor at the zero wavevector into the non-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 the absence of the interaction-induced restoring force for the short-range dynamics (i.e., eqn (87)).

Combining eqn (76) and the dynamical discussion in Section VIII suggests that the above-mentioned second scenario for the 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-range-correlated φ-field; for it seems plausible that the long-range-correlated potential field arises from the elastic nature of the contact network. From the discussion, we infer that the virial-type 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 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 can be extended to address the non-hyperuniform behaviors of other measures than the density–density structure factor, which are obtained from various physical quantities including the local number variance for a window^{17} and the contact number fluctuations.^{102,103} We also envision that advancing stochastic DFT^{44–56} at strong coupling will pave the way for the realistic description of quasi-localized soft modes induced by the intermittent rearrangements of particles such as bucklers.^{19–22,76–83}

(A1) |

(A2) |

(A3) |

(A4) |

ψ(r) = ϕ(r) + iψ_{dft}(r), | (A5) |

(A6) |

〈_{N}(r)〉_{eq} = ρ(r). | (A7) |

For later convenience, we also introduce the intrinsic Helmholtz free energy F[ρ, 0], the central functional of equilibrium DFT:^{58–61}

(A8) |

(A9) |

(A10) |

(A11) |

(A12) |

(A13) |

(A14) |

(A15) |

(A16) |

(A17) |

(A18) |

In eqn (A17), we use the following approximation:

(A19) |

(A20) |

(A21) |

(B1) |

(B2) |

(B3) |

(B4) |

(B5) |

(B6) |

(B7) |

(B8) |

(B9) |

(B10) |

(C1) |

(C2) |

(C3) |

(C4) |

(C5) |

Before proceeding to the ϕ-field averaging operation defined by eqn (17), we clarify the corresponding terms where the strong-coupling approximation needs to be developed. To this end, let ζ[] be the general functional given by the sum of the -independent part ζ_{c} and the remaining -dependent contribution ζ_{r}[]:

ζ[] = ζ_{c} + ζ_{r}[]. | (C6) |

(C7) |

(C8) |

(C9) |

(C10) |

(C11) |

(C12) |

(C13) |

(C14) |

(C15) |

In the remaining subsections, we will evaluate eqn (74) and (82) based on the above strong-coupling approximation represented by eqn (C14).

(C16) |

(C17) |

(C18) |

First, the Gaussian integration over the φ-field yields

(C19) |

Next, we investigate the third term on the rhs of eqn (C14), or . The expression (C4) of implies the necessity of evaluating the following contribution:

(C20) |

(C21) |

(C22) |

(C23) |

(C24) |

(C25) |

(C26) |

(C27) |

(C28) |

(C29) |

(C30) |

(C31) |

(C32) |

Substituting eqn (C30)–(C32) into eqn (C29), we obtain

(C33) |

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