Equilibrium States Corresponding to Targeted Hyperuniform Nonequilibrium Pair Statistics Haina Wang1and Salvatore Torquato2 3a

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Equilibrium States Corresponding to Targeted Hyperuniform Nonequilibrium Pair

Statistics

Haina Wang1and Salvatore Torquato2, 3, a)

1)Department of Chemistry, Princeton University, Princeton, New Jersey, 08544,

USA

2)Department of Chemistry, Department of Physics, Princeton Institute of Materials,

and Program in Applied and Computational Mathematics, Princeton University,

Princeton, New Jersey 08544, USA

3)School of Natural Sciences, Institute for Advanced Study, 1 Einstein Drive,

Princeton, NJ 08540, USA

arXiv:2210.06446v3 [cond-mat.soft] 19 Dec 2022

The Zhang-Torquato conjecture [Phys. Rev. E 101, 032124 (2020)] states that any

realizable pair correlation function g2(r) or structure factor S(k) of a translation-

ally invariant nonequilibrium system can be attained by an equilibrium ensemble

involving only (up to) eﬀective two-body interactions. To further test and study this

conjecture, we consider two singular nonequilibrium models of recent interest that

also have the exotic hyperuniformity property: a 2D “perfect glass” and a 3D criti-

cal absorbing-state model. We ﬁnd that each nonequilibrium target can be achieved

accurately by equilibrium states with eﬀective one- and two-body potentials, lending

further support to the conjecture. To characterize the structural degeneracy of such

a nonequilibrium-equilibrium correspondence, we compute higher-order statistics for

both models, as well as those for a hyperuniform 3D uniformly randomized lattice

(URL), whose higher-order statistics can be very precisely ascertained. Interestingly,

we ﬁnd that the diﬀerences in the higher-order statistics between nonequilibrium

and equilibrium systems with matching pair statistics, as measured by the “hole”

probability distribution, provides measures of the degree to which a system is out

of equilibrium. We show that all three systems studied possess the bounded-hole

property, and that holes near the maximum hole size in the URL are much rarer

than those in the underlying simple cubic lattice. Remarkably, upon quenching,

the eﬀective potentials for all three systems possess local energy minima (i.e., in-

herent structures) with stronger forms of hyperuniformity compared to their target

counterparts. Our methods are expected to facilitate the self-assembly of tunable

hyperuniform soft-matter systems.

a)Email: torquato@princeton.edu

I. INTRODUCTION

Probing and characterizing structural properties of many-body systems in and out of

equilibrium is a crucial task in the understanding of a large variety of physical, chemical

and biological phenomena.1–5An outstanding challenge is the determination of eﬀective

interactions in many-body systems that accurately yield equilibrium states with prescribed

pair statistics. Solving such inverse problems is a powerful way to tackle the unsolved

problem concerning the realizability of prescribed functional forms of pair statistics by many-

body systems.6–11 Such investigations also enable one to probe systems with identical pair

statistics but diﬀerent higher-body statistics, which is expected to shed light on the well-

known degeneracy problem of statistical mechanics.12–14 Moreover, such eﬀective potentials

can be used to model macromolecules and solutions,15,16 and to design nanoparticles that

self-assemble into desired structures, thereby facilitating material discovery.17–20

Recently, Zhang and Torquato conjectured that any realizable pair correlation function

g2(r) or structure factor S(k) corresponding to a translationally invariant nonequilibrium

system can be attained by an equilibrium ensemble involving only one- and two-body ef-

fective interactions at positive temperatures.11 Testing the conjecture requires the precise

determination of the eﬀective interactions for a spectrum of target systems, including those

with the exotic hyperuniform property.21,22 Disordered hyperuniform many-body systems,

which can be solid and ﬂuid states, are unusual amorphous states of matter that lie be-

tween a crystal and liquid. They are like perfect crystals in the way they suppress large-

scale density ﬂuctuations, and yet are like liquids or glasses in that they are statistically

isotropic with no Bragg peaks;21,22 see Sec. II B for precise deﬁnitions. Disordered hype-

runiform states play vital roles in a variety of diﬀerent contexts. For example, disordered

stealthy ground states have been discovered23–25 corresponding to under soft, long-ranged

interactions, which are highly degenerate and have the “bounded-hole” property, which is

a singular characteristic for a disordered system.26,27 Network structures derived from dis-

ordered stealthy point patterns can achieve complete photonic band gaps and have novel

optical properties that were previously not thought to be possible.28,29 Disordered hype-

runiform states also arise in the eigenvalues of random matrices (such as the Gaussian

unitary ensemble),30–32 ground states of free fermions,31 as well as one-component plas-

mas at positive temperatures,33–35 all of which are exotic ﬂuid states in which the par-

ticles interact Coulombically.36 Other examples include including glass formation,22,37,38

jamming,39–43 rigidity,38,44 biology,45,46 localization of waves and excitations,47–50 antenna

or laser array designs,51 self-organization,52–54 ﬂuid dynamics,55–57 quantum systems,31,58–61

and pure mathematics.62–66 Because disordered hyperuniform states combine the advantages

of statistical isotropy and the suppression of density ﬂuctuations on large scales, they can

be endowed with novel physical properties.22,28,29,37,38,45,46,52–54,67

While Zhang and Torquato introduced an algorithm to draw equilibrium classical parti-

cle conﬁgurations from canonical ensembles with one- and two-body interactions that cor-

respond to targeted functional forms for g2(r) or S(k), the algorithm does not generate

explicit forms of the potentials.11 Very recently, Torquato and Wang developed an inverse

methodology that determines eﬀective interactions with unprecedented accuracy.68 Using

this procedure, they demonstrated the realizability of g2(r) for all rand S(k) for all kfor

two diﬀerent nonequilibrium models, including a two-dimensional (2D) nonhyperuniform

random sequential addition process and a 3D hyperuniform “cloaked” uniformly random-

ized lattice (URL).68 However, the Zhang-Torquato conjecture remains largely untested.

In this paper, we utilize this precise inverse methodology68 to further test and study the

Zhang-Torquato conjecture for unusual nonequilibrium hyperuniform systems. Hyperuni-

form targets are particularly challenging because at positive temperature Tthey require a

long-ranged pair interaction v(r) that must be balanced by one-body potentials to stabilize

the equilibrium system.22

Prior to the development of our inverse methodology,68 predictor-corrector methods,69–72

such as Iterative Boltzmann inversion (IBI)71 and iterative hypernetted chain inversion

(IHNCI),72,73 were regarded to be the most accurate inverse procedures. Both IBI and

IHNCI begin with an initial discretized (binned) approximation of a trial pair potential.

The trial pair potential at each binned distance is iteratively updated to attempt to reduce

the diﬀerence between the target and trial pair statistics. However, IBI and IHNCI cannot

treat long-ranged pair interactions required for hyperuniform targets, nor do they consider

one-body interactions that stabilize hyperuniform equilibrium states;68 see Sec. II B for

details. These algorithms also accumulate random errors in the binned potentials due to

simulation errors in the trial pair statistics, and thus do not achieve the precision required

to probe realizability problems. Moreover, because all previous methods do not optimize a

pair-statistic “distance” functional, they are unable to detect poor agreement between the

target and trial pair statistics that may arise as the simulation evolves, leading to increas-

ingly inaccurate corresponding trial potentials, as demonstrated in Ref. 68.

The inverse methodology presented in Ref. 68 improves on previous procedures in several

signiﬁcant ways. It utilizes a parameterized family of pointwise basis functions for the

potential function at T > 0, whose initial form is informed by small- and large-distance

behaviors dictated by statistical-mechanical theory. Pointwise potential functions do not

suﬀer from the accumulation of random errors during a simulation, resulting in more accurate

interactions.68 Since it has recently been established74 that inverse methods that target

only g2(r) or only S(k) for a limited range of ror kmay generate eﬀective potentials

that are distinctly diﬀerent from the unique potential dictated by Henderson’s theorem,75

our methodology68 minimizes an objective function that incorporates both the target pair

correlation function g2(r) and structure factor S(k) so that both the small- and large-distance

correlations are very accurately captured. For hyperuniform targets, our methodology is able

to optimize the required long-ranged pair potential76 as well as the neutralizing background

one-body potential;68 see Sec. IV for details.

To assess the accuracy of inverse methodologies to target pair statistics, we introduced68

the following dimensionless L2-norm error:

E=pDg2+DS,(1)

where Dg2and DSare L2functions, given by

Dg2=ρZRd

[g2,T (r)−g2,F (r;a)]2dr,(2)

DS=1

ρ(2π)dZRd

[ST(k)−SF(k;a)]2dk,(3)

where g2,F (r;a) and SF(k;a) represent the ﬁnal pair statistics at the end of the optimization,

which depend on the vector of potential parameters a. We have previously shown that our

method is able to treat challenging near-critical and hyperuniform targets,68 which previous

methods cannot do. Thus, it is the only available method to determine eﬀective interactions

for nonequilibrium hyperuniform pair statistics. Moreover, in cases where IBI and IHNCI

are able to achieve optimized potentials, e.g., for equilibrium target pair statistics without

long-range interactions, our inverse methodology generally yields L2-norm errors (1) that are

an order of magnitude smaller than those via previous methods, and reaches the precision

required to recover the unique potential dictated by Henderson’s theorem.75

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EquilibriumStatesCorrespondingtoTargetedHyperuniformNonequilibriumPairStatisticsHainaWang1andSalvatoreTorquato2,3,a)1)DepartmentofChemistry,PrincetonUniversity,Princeton,NewJersey,08544,USA2)DepartmentofChemistry,DepartmentofPhysics,PrincetonInstituteofMaterials,andPrograminAppliedandComputationalMa...

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Equilibrium States Corresponding to Targeted Hyperuniform Nonequilibrium Pair Statistics Haina Wang1and Salvatore Torquato2 3a

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