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REVIEW 4 major objections 5 minor 152 references

Three-Dimensional Construction of Hyperuniform, Nonhyperuniform and Antihyperuniform Random Media via Spectral Density Functions and Their Transport Properties

T0 review · 4 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper reports the first explicit construction of three-dimensional antihyperuniform two-phase media, together with hyperuniform and nonhyperuniform realizations, all generated from prescribed spectral density functions whose transport…

desk verdict Plausible 3D extension of the group's 2D spectral-density construction, with a first-claim for antihyperuniform media that needs direct variance-scaling evidence before it is fully established. read the letter →

arxiv 2412.08974 v1 pith:QOCWDEO2 submitted 2024-12-12 cond-mat.mtrl-sci cond-mat.stat-mech

Wenlong Shi , Yang Jiao , Salvatore Torquato This is my paper · ORCID
classification cond-mat.mtrl-scicond-mat.stat-mech
keywords hyperuniformityantihyperuniformmediaspectraldensityfunctionmicrostructurereconstructiondiffusionspreadabilityfluidpermeabilitytwo-phaserandomsimulatedannealing
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper establishes a computational route from a prescribed spectral density function to a three-dimensional two-phase random microstructure, and shows that the transport properties of the result can be read off from the same function. It constructs disordered realizations in four classes—stealthy hyperuniform, standard hyperuniform, nonhyperuniform, and antihyperuniform—at several volume fractions and correlation lengths, including what it reports as the first explicit three-dimensional antihyperuniform two-phase media. The paper then computes the diffusion spreadability exactly and estimates fluid permeability from the target spectral functions, finding that the length-scale parameter alone can change spreadability by orders of magnitude at intermediate and long times while leaving the asymptotic decay class unchanged. If correct, the method turns a small set of analytic autocovariance models into a practical inverse-design tool for composites with targeted diffusive and fluid-transport behavior.

What carries the argument

The spectral density function $\tilde{\chi}_V(k)$, the Fourier transform of the autocovariance $\chi_V(r)=S_2^{(i)}(r)-\phi_i^2$, is the object that carries the argument. Because both the diffusion spreadability and the fluid-permeability estimate are expressible directly in terms of $\tilde{\chi}_V(k)$, targeting $\tilde{\chi}_V(k)$ couples microstructure generation to transport prediction without intermediate reconstruction steps. The construction itself is driven by an energy functional $E=\sum_{k<K^*}(\tilde{\chi}_V(k)-\tilde{\chi}_V^0(k))^2$ minimized by simulated annealing, with the generalized collective coordinate $\tilde{J}(k)$ updated locally after each voxel swap so that the spectral density of a trial configuration is obtained in $O(1)$ time per move. The four analytic families of target functions supply the length-scale parameter $a$ that tunes morphology within each class.

What would settle it

Measure the spectral density of the constructed antihyperuniform samples at the smallest resolved wavenumbers, or compute the local volume-fraction variance in large spherical windows, and check for the predicted $\tilde{\chi}_V(k)\sim 2\pi^2/k$ divergence and for a variance decay slower than $R^{-3}$; if the small-$k$ spectrum saturates to a finite value because of the $L=128$ cutoff, the claimed first realization of true antihyperuniform media in 3D would not be supported by the constructed finite systems.

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Extended reading notes

Core claim

On its own terms, the central discovery is that a Fourier-space simulated-annealing procedure can realize explicit three-dimensional two-phase media for analytic spectral densities spanning every hyperuniformity class, including the previously unrealized antihyperuniform case. The construction minimizes the squared difference between the angular-averaged spectral density of a binary voxel configuration and a target function $\tilde{\chi}_V(k)$, using rapidly updated collective coordinates to make each trial move cheap. With targets given by the Debye exponential, the damped-oscillatory hyperuniform model, the stealthy zero-region model, and the power-law autocovariance model $\chi_V(r;a)=\phi_1\phi_2/[1+2(r/a)+(r/a)^2]$, the paper produces media at $\phi_1=0.25,0.5,0.75$ and $a=5,10,25$ voxels. It reports that the antihyperuniform media contain clusters of dramatically different sizes and morphologies, mimicking critical-point fluctuations. Using the exact spreadability formula and a reference-based permeability estimate, it finds asymptotic decay exponents consistent with theory and an ordering in which antihyperuniform media have the largest dimensionless permeability $k/a^2$ and stealthy hyperuniform media the smallest.

Load-bearing premise

The load-bearing premise is that the analytic autocovariance functions chosen as targets, in particular the power-law antihyperuniform model of Eq. (46), are genuinely realizable as three-dimensional two-phase media and that a finite $L=128$ voxel construction reproduces the diverging small-wavenumber behavior of Eq. (47); the paper notes that sufficient realizability conditions remain an open problem.

Editorial extensions

If this is right

  • Any spectral density that satisfies the known necessary conditions can now be converted into an explicit 3D two-phase microstructure, including antihyperuniform media that had only been treated analytically.
  • Because spreadability and permeability are computed directly from the target $\tilde{\chi}_V(k)$, the length-scale parameter $a$ and volume fraction $\phi_1$ can be optimized for a desired transport property before any microstructure is generated.
  • Within a fixed hyperuniformity class, changing $a$ changes the excess spreadability by orders of magnitude at intermediate and long times, so the asymptotic decay exponent alone does not determine the transport efficiency.
  • The dimensionless permeability $k/a^2$ decreases with increasing $\phi_1$ and increasing $a$, and among the four classes antihyperuniform media give the largest $k/a^2$ while stealthy hyperuniform media give the smallest.
  • The same Fourier-space procedure can be adapted to non-isotropic target spectra and to dynamic wave properties, extending the 3D construction beyond diffusive transport.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A direct test the paper does not perform is to measure the local volume-fraction variance of the constructed antihyperuniform media; if the finite-size samples fail to show the slow $R^{-2}$ decay, the realization claim would need to be restricted to finite systems.
  • The permeability estimates inherit the accuracy of the reference-sample scaling in Eq. (28), so the reported ordering of $k/a^2$ across classes should be checked against direct pore-scale flow simulations before being used for design.
  • Since sufficient realizability conditions are open, success at one resolution does not guarantee that the same target function is realizable at other resolutions or volume fractions; the construction may be selecting a particular finite-size representative of a larger equivalence class.
  • The same construction machinery could be applied to mixtures of basis spectral densities to engineer hybrid microstructures, a direction the paper mentions only as a possibility for parameterized targets.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper presents a Fourier-space simulated-annealing method for constructing three-dimensional, statistically isotropic two-phase random media from prescribed analytical spectral density functions. The target functions cover Debye random media (Eqs. 39-40), standard hyperuniform media (Eqs. 42-43), stealthy hyperuniform media (Eq. 45), and antihyperuniform media with a power-law autocovariance (Eqs. 46-47), for volume fractions 0.25, 0.5, and 0.75 and correlation lengths a=5, 10, and 25 voxels. It claims the first realization of antihyperuniform two-phase media in 3D, supported by visual renderings showing clusters of very different sizes. It then computes the diffusion spreadability from Eq. (20) and estimates fluid permeability from Eq. (28) directly from the target spectral density functions, reporting that varying the length-scale parameter produces orders-of-magnitude changes in spreadability and that antihyperuniform media have the largest dimensionless permeability.

Significance. If fully verified, the framework would usefully extend the Yeong-Torquato Fourier-space construction approach to 3D and provide a parameterized route to microstructures across all hyperuniformity classes, including a candidate antihyperuniform medium. The computational strategy is efficient, the phase-inversion symmetry of the constructed media is a sensible consistency check, and the reported long-time spreadability scalings agree with known theory. However, the central advertised novelty, the first 3D antihyperuniform two-phase media, is not yet established because the paper never measures the spectral density or local volume-fraction variance of the constructed arrays; the transport results are also properties of the target functions rather than of the constructed realizations. These gaps are fixable and do not invalidate the method, but they must be addressed before the central claims can be accepted.

major comments (4)
  1. [Sec. IV, Figs. 1-8; Eq. (17)] The constructed media are never quantitatively verified against their targets. The paper reports no measured angular-averaged spectral density of the final arrays and no local volume-fraction variance sigma_V^2(R). For the antihyperuniform claim, Eq. (17) requires sigma_V^2(R) ~ R^{-(d+alpha)} = R^{-2} in d=3 for alpha=-1, and Eq. (50) requires a diverging spectral density ~1/k. A finite periodic binary array has exactly zero spectral density at k=0 and only discrete wavevectors down to 2*pi/L, so matching the target at the constrained shells in Eq. (35) does not by itself establish the infinite-volume divergence. Please show measured tilde-chi_V(k) from the final arrays for all four classes, and sigma_V^2(R) scaling for the antihyperuniform case, including at least one larger system size.
  2. [Eq. (48)] The definitions of Ci and Si in Eq. (48) are incorrect. As printed, Ci(x) = integral_0^x cos(t)/t dt and Si(x) = integral_0^x cos(t)/t dt are identical, and the Ci integrand is not integrable at t=0; Si should contain sin(t), and Ci is conventionally defined with the lower limit at infinity. Since Eq. (47) is the target spectral density for the central antihyperuniform construction, the formula must be corrected and the construction checked against the corrected expression. If the code actually used standard definitions, this should be stated explicitly.
  3. [Sec. V, Eqs. (20) and (28)] The spreadability and permeability results are evaluated directly from the parameterized target functions, not from the constructed voxel arrays. Consequently, the observed trends, including the orders-of-magnitude variation of S(t) with a and the permeability ranking among the four classes, are inherited from the input spectral densities and do not test the construction. The abstract and title present these as properties of the constructed media. Either compute S(t) and permeability (or at least l_p^2) from the realizations themselves, or explicitly reframe Section V as analytical predictions for the target spectral-density models rather than for the constructed microstructures.
  4. [Sec. III.A; Sec. II.A] The underdetermined nature of the inverse problem is acknowledged in Sec. III.A (N_Omega < N, no unique solution to Eq. (33)), and Sec. II.A notes that sufficient realizability conditions for autocovariance functions are an open problem. These limitations are particularly important for the singular antihyperuniform target: many binary arrays can satisfy the finite set of discrete-k constraints without exhibiting the required R^{-2} variance scaling. The paper states that L=64 and L=256 were investigated to verify resolution effects, but no results are shown. Please include this finite-size convergence data and the variance scaling evidence.
minor comments (5)
  1. [Table I] The entry 0.09539 for the standard hyperuniform medium at phi1=0.5 and a=10 is an apparent outlier relative to the neighboring entries 0.9603 (a=5) and 0.9218 (a=25); please check this value and the associated calculation.
  2. [Sec. IV.A] The phrase 'Interesting, we note' should read 'Interestingly, we note.'
  3. [Sec. IV.D, last paragraph] The sentence describing 'density fluctuations that strongly suppress scattering' appears to be a typo: for antihyperuniform media the zero-wavenumber scattering diverges, so the fluctuations should enhance, not suppress, scattering.
  4. [Fig. 9] Please specify whether the horizontal axis is time t or the dimensionless Dt/a^2. If Dt/a^2 is used, the claim of orders-of-magnitude variation with a needs to be justified beyond the explicit a^2 rescaling in Eq. (22).
  5. [Sec. III.A, Eq. (32)] The text and Eq. (32) refer to 'pixel' and a 'square pixel,' but the construction is three-dimensional; the terminology should be voxel and cubic voxel.

Circularity Check

2 steps flagged · score 6.0 of 10

Transport properties are integrals of the target spectral densities used as construction inputs, making the S(t) and permeability 'results' inherit the input functions by construction.

  1. self definitional [Sec. V (Transport Properties of Constructed Media), Eqs. (20), (27), (28); Abstract]
    "We also determine the diffusion spreadability S(t) and estimate the fluid permeability k associated with all of the constructed materials directly from the corresponding χ̃V(k) functions. ... Since both the diffusion spreadability and fluid permeability are directly obtained from the parameterized function χ̃V(k; a), these quantities also explicitly depend on the length scale parameter a."

    Eq. (20) defines S(t) as the Fourier-space integral of χ̃V, and Eq. (28) estimates k from ℓ_p^2, which Eq. (27) computes from the same χ̃V. The χ̃V used in both evaluations is the analytic target function prescribed before construction, not the spectral density measured from any realized voxel array. Therefore the reported orders-of-magnitude variation of S(t) with a, and the permeability ordering across the four classes, are properties of the chosen input functions by construction. The paper is transparent about computing 'directly from the corresponding χ̃V(k) functions,' but labeling these as properties of 'the constructed materials' closes an input-output loop without using the constructed microstructures.

  2. other [Sec. V, paragraph following Fig. 9; echoed in Sec. VI]
    "Torquato [22] showed that the long-time excess spreadability for the above media possesses the following scaling behavior: ∼ t−3/2 for Debye random media, ∼ t−5/2 for standard hyperuniform media, ∼ e−θt/t for stealthy hyperuniform media, and ∼ t−1 for antihyperuniform media, which are in complete agreement with our numerical results."

    The numerical results are produced by applying Eq. (20), the same exact spreadability formula from which Ref. [22] derived those long-time scalings. Feeding the same analytic χ̃V targets into the same integral transform cannot test the cited theory; the agreement is tautological. The self-citation is used as if it were an external prediction, but the computation and the cited scaling law share the same input-to-output mapping, so the confirmation carries no independent weight and does not validate the realized microstructures.

full rationale

The construction pipeline itself is not circular: the simulated-annealing energy in Eqs. (34)-(35) independently optimizes voxel configurations against a prescribed χ̃V, and the Debye and standard-hyperuniform targets are standard benchmark models. The circular reduction is confined to the transport section, where S(t) and k are evaluated from the analytic target functions rather than from the realized arrays; the resulting trends are therefore inherited from the inputs, not emergent from the generated microstructures. The separate claim of a 'first realization of antihyperuniform two-phase materials in 3D' is not circular, because it rests on the actual simulated-annealing output, but it is under-verified: the paper never reports the measured χ̃V or local volume-fraction variance σ_V^2(R) of the constructed arrays, so the required R^{-(d+α)} scaling (Eq. 17 with α=-1) is not demonstrated. That is a correctness/verification gap rather than a circularity. The self-referential confirmation of Ref. [22] adds a minor circular flavor. Overall score 6: one class of reported results reduces by construction to the input spectral densities, while the core construction claim retains independent computational content.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

No new physical entities are introduced. The central outputs depend on the chosen length scale a and volume fraction phi1 as inputs, on prior realizability claims for the autocovariance models, and on two transport-formula approximations from Refs. 22 and 25.

free parameters (3)
  • correlation length a = 5, 10, 25 voxels
    Length-scale parameter in each target spectral density function; chosen by hand to explore microstructure families, not fitted to data. All reported transport trends are functions of a.
  • phase volume fraction phi1 = 0.25, 0.5, 0.75
    Volume fraction of phase 1, set as input for construction; affects percolation and all reported transport values.
  • hyperuniform oscillation constraint (qa)^2 = 1/3
    Chosen to make Eq. (42) satisfy the hyperuniformity sum rule; a constraint rather than a fit, but the specific value determines the family of standard hyperuniform media.
assumptions (4)
  • domain assumption Target autocovariance functions Eqs. (39), (42), and (46) satisfy all known necessary conditions on two-point correlation functions of two-phase media.
    Stated in Sec. IV; sufficiency of these conditions remains an open problem (Sec. II.A), so realizations are not guaranteed a priori. The antihyperuniform model relies on Torquato 2016 (Ref. 139).
  • domain assumption The permeability approximation Eq. (28) with a simple-cubic hard-sphere reference is accurate for the constructed media.
    Adopted from Ref. 25; the paper notes it is an estimate, and no direct flow simulation is performed.
  • standard math The diffusion spreadability formula Eq. (20) applies to the constructed digitized media with periodic boundary conditions.
    Exact for two-phase media given chi_tilde; used directly with the analytic target functions rather than the realized voxel arrays.
  • domain assumption Finite-size L=128 voxel realizations faithfully represent the infinite-system spectral density behavior.
    The paper states checks at L=64 and L=256 but provides no quantitative data; especially for the diverging small-k antihyperuniform spectrum, finite-size truncation is a concern.

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Pith. "Pith review of Three-Dimensional Construction of Hyperuniform, Nonhyperuniform and Antihyperuniform Random Media via Spectral Density Functions and Their Transport Properties." pith.science (2026). https://pith.science/paper/QOCWDEO2

@misc{pith2026241208974,
  author       = {Pith},
  title        = {Pith review of: Three-Dimensional Construction of Hyperuniform, Nonhyperuniform and Antihyperuniform Random Media via Spectral Density Functions and Their Transport Properties},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QOCWDEO2}},
  note         = {Machine review of arXiv:2412.08974}
}
abstract

Rigorous theories connecting physical properties of a heterogeneous material to its microstructure offer a promising avenue to guide the computational material design and optimization. We present here an efficient Fourier-space based computational framework and employ a variety of analytical ${\tilde \chi}_{_V}({k})$ functions that satisfy all known necessary conditions to construct 3D disordered stealthy hyperuniform, standard hyperuniform, nonhyperuniform, and antihyperuniform two-phase heterogeneous material systems at varying phase volume fractions. We show that a rich spectrum of distinct structures within each of the above classes of materials can be generated by tuning correlations in the system across length scales. We present the first realization of antihyperuniform two-phase heterogeneous materials in 3D, which are characterized by a power-law autocovariance function $\chi_{_V}(r)$ and contain clusters of dramatically different sizes and morphologies. We also determine the diffusion spreadability ${\cal S}(t)$ and estimate the fluid permeability $k$ associated with all of the constructed materials directly from the corresponding ${\tilde \chi}_{_V}({k})$ functions. We find that varying the length-scale parameter within each class of ${\tilde \chi}_{_V}({k})$ functions can also lead to orders of magnitude variation of ${\cal S}(t)$ at intermediate and long time scales. Moreover, we find that increasing solid volume fraction $\phi_1$ and correlation length $a$ in the constructed media generally leads to a decrease in the dimensionless fluid permeability $k/a^2$. These results indicate the feasibility of employing parameterized ${\tilde \chi}_{_V}({k})$ for designing composites with targeted transport properties.

Figures

Figures reproduced from arXiv: 2412.08974 by the authors.

Figure 2
Figure 2. FIG. 2. Constructions of the Debye random media with [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 1
Figure 1. FIG. 1. Rescaled autocovariance function [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Rescaled autocovariance function [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Constructions of the standard hyperuniform media [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Rescaled autocovariance function [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Constructions of the stealthy hyperuniform media [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Constructions of the antihyperuniform media with [PITH_FULL_IMAGE:figures/full_fig_p010_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Excess spreadability [PITH_FULL_IMAGE:figures/full_fig_p011_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Constructed microstructures with [PITH_FULL_IMAGE:figures/full_fig_p013_10.png]

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    Ultra-efficient reconstruction of 3d microstructure and distribution of properties of random heterogeneous materials con- taining multiple phases,

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    Hierarchical n-point polytope func- tions for quantitative representation of complex hetero- geneous materials and microstructural evolution,

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    Accurate modeling and re- construction of three-dimensional percolating filamen- tary microstructures from two-dimensional micrographs via dilation-erosion method,

    En-Yu Guo, Nikhilesh Chawla, Tao Jing, Salvatore Torquato, and Yang Jiao, “Accurate modeling and re- construction of three-dimensional percolating filamen- tary microstructures from two-dimensional micrographs via dilation-erosion method,” Materials Characteriza- tion 89, 33–42 (2014)

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    Designing disordered hype- runiform two-phase materials with novel physical prop- erties,

    D. Chen and S. Torquato, “Designing disordered hype- runiform two-phase materials with novel physical prop- erties,” Acta Mater. 142, 152–161 (2018)

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    Computational design of anisotropic stealthy hyperuniform composites with en- gineered directional scattering properties,

    Wenlong Shi, David Keeney, Duyu Chen, Yang Jiao, and Salvatore Torquato, “Computational design of anisotropic stealthy hyperuniform composites with en- gineered directional scattering properties,” Physical Re- view E 108, 045306 (2023)

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    Local density fluctu- ations, hyperuniformity, and order metrics,

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    Hyperuniformity in point patterns and two-phase random heterogeneous media,

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    Hyperuniformity and its generalizations,

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    Hyperuniform states of matter,

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    Designer disordered materials with large, com- plete photonic band gaps,

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    Photonic band gap in isotropic hyper- uniform disordered solids with low dielectric contrast,

    Weining Man, Marian Florescu, Kazue Matsuyama, Polin Yadak, Geev Nahal, Seyed Hashemizad, Eric Williamson, Paul Steinhardt, Salvatore Torquato, and Paul Chaikin, “Photonic band gap in isotropic hyper- uniform disordered solids with low dielectric contrast,” Optics express 21, ...

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    Isotropic band gaps and freeform waveguides observed in hyperuniform dis- ordered photonic solids,

    Weining Man, Marian Florescu, Eric Paul Williamson, Yingquan He, Seyed Reza Hashemizad, Brian YC Le- ung, Devin Robert Liner, Salvatore Torquato, Paul M Chaikin, and Paul J Steinhardt, “Isotropic band gaps and freeform waveguides observed in hyperuniform dis- ordered photonic ...

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    Evolving scattering networks for engineer- ing disorder,

    Sunkyu Yu, “Evolving scattering networks for engineer- ing disorder,” Nat. Comput. Sci. 1, 1 (2023)

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    Hearing the shape of a drum for light: isospectrality in photonics,

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    Wave propagation and band tails of two- dimensional disordered systems in the thermodynamic limit,

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    Over 65% sunlight absorption in a 1 µm si slab with hyper- uniform texture,

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    Engineered disorder in photonics,

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    Biological tissue- 16 inspired tunable photonic fluid,

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    Transport, geometrical, and topological properties of stealthy dis- ordered hyperuniform two-phase systems,

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    Characterization of void space, large- scale structure, and transport properties of maximally random jammed packings of superballs,

    Charles Emmett Maher, Frank H Stillinger, and Salva- tore Torquato, “Characterization of void space, large- scale structure, and transport properties of maximally random jammed packings of superballs,” Physical Re- view Materials 6, 025603 (2022)

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    Microstructure and mechan- ical properties of hyperuniform heterogeneous materi- als,

    Yaopengxiao Xu, Shaohua Chen, Pei-En Chen, Wenxi- ang Xu, and Yang Jiao, “Microstructure and mechan- ical properties of hyperuniform heterogeneous materi- als,” Physical Review E 96, 043301 (2017)

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    Multifunctional hyperuniform cellular networks: optimality, anisotropy and disorder,

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    Multifunctional composites for elastic and electromagnetic wave prop- agation,

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    Extraordinary disordered hyper- uniform multifunctional composites,

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    Unexpected density fluctuations in jammed disordered sphere packings,

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    Hyperuniform long-range correlations are a signature of disordered jammed hard-particle packings,

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    Maximally random jammed packings of platonic solids: Hyperuniform long- range correlations and isostaticity,

    Yang Jiao and Salvatore Torquato, “Maximally random jammed packings of platonic solids: Hyperuniform long- range correlations and isostaticity,” Physical Review E 84, 041309 (2011)

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    Equi- librium phase behavior and maximally random jammed state of truncated tetrahedra,

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    Suppressed com- pressibility at large scale in jammed packings of size- disperse spheres,

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    Diagnosing hy- peruniformity in two-dimensional, disordered, jammed packings of soft spheres,

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    Hyperuniformity of critical absorbing states,

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    Hyperuniformity and phase separation in biased en- sembles of trajectories for diffusive systems,

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    Emergent hyperuniformity in period- ically driven emulsions,

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    Hyperuniform density fluctuations and diverging dynamic correlations in periodically driven colloidal suspensions,

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    Hyperuniform monocrystalline structures by spinodal solid-state dewetting,

    Marco Salvalaglio, Mohammed Bouabdellaoui, Monica Bollani, Abdennacer Benali, Luc Favre, Jean-Benoit Claude, Jerome Wenger, Pietro de Anna, Francesca In- tonti, Axel Voigt,et al., “Hyperuniform monocrystalline structures by spinodal solid-state dewetting,” Physical Review Lett...

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    Noise, diffusion, and hyperuniformity,

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    En- hanced hyperuniformity from random reorganization,

    Daniel Hexner, Paul M Chaikin, and Dov Levine, “En- hanced hyperuniformity from random reorganization,” Proceedings of the National Academy of Sciences 114, 4294–4299 (2017)

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    Mixing by unstirring: hyperuniform dispersion of interacting particles upon chaotic advection,

    Joost H Weijs and Denis Bartolo, “Mixing by unstirring: hyperuniform dispersion of interacting particles upon chaotic advection,” Physical review letters 119, 048002 (2017)

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    Nonequilibrium strongly hyperuniform fluids of circle active particles with large local density fluctuations,

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    Random-organizing hyper- uniform fluids with momentum-conserved activations,

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    Anomalous local coordination, density fluctuations, and void statis- tics in disordered hyperuniform many-particle ground states,

    Chase E Zachary and Salvatore Torquato, “Anomalous local coordination, density fluctuations, and void statis- tics in disordered hyperuniform many-particle ground states,” Physical Review E 83, 051133 (2011)

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    Ensemble theory for stealthy hyperuniform disordered ground states,

    Salvatore Torquato, G Zhang, and FH Stillinger, “Ensemble theory for stealthy hyperuniform disordered ground states,” Physical Review X 5, 021020 (2015)

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    Constraints on collective density variables: Two dimensions,

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    Classical disordered ground states: Super- ideal gases and stealth and equi-luminous materials,

    Robert D Batten, Frank H Stillinger, and Salvatore Torquato, “Classical disordered ground states: Super- ideal gases and stealth and equi-luminous materials,” Journal of Applied Physics 104, 033504 (2008)

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    Novel low-temperature behavior in classi- cal many-particle systems,

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    Charge fluctuations in coulomb sys- tems,

    Joel L Lebowitz, “Charge fluctuations in coulomb sys- tems,” Physical Review A 27, 1491 (1983)

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    Ground states of stealthy hyperuniform potentials: I. entropically favored configurations,

    Ge Zhang, Frank H Stillinger, and Salvatore Torquato, “Ground states of stealthy hyperuniform potentials: I. entropically favored configurations,” Physical Review E 92, 022119 (2015)

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    Ground states of stealthy hyperuniform potentials. ii. stacked-slider phases,

    Ge Zhang, Frank H Stillinger, and Salvatore Torquato, 17 “Ground states of stealthy hyperuniform potentials. ii. stacked-slider phases,” Physical Review E 92, 022120 (2015)

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    Point processes in arbitrary dimension from fermionic gases, random matrix theory, and number the- ory,

    Salvatore Torquato, A Scardicchio, and Chase E Zachary, “Point processes in arbitrary dimension from fermionic gases, random matrix theory, and number the- ory,” Journal of Statistical Mechanics: Theory and Ex- periment 2008, P11019 (2008)

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    Disordered hyperuniform vortex matter with rhombic distortions in fese at low fields,

    Jazm ´ ın Arag´ on S´ anchez, Ra´ ul Cort´ es Maldonado, M Lourdes Amig´ o, Gladys Nieva, Alejandro Kolton, and Yanina Fasano, “Disordered hyperuniform vortex matter with rhombic distortions in fese at low fields,” Physical Review B 107, 094508 (2023)

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    Nearly hyperuniform network models of amorphous silicon,

    Miroslav Hejna, Paul J Steinhardt, and Salvatore Torquato, “Nearly hyperuniform network models of amorphous silicon,” Physical Review B 87, 245204 (2013)

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    Hy- peruniformity in amorphous silicon based on the mea- surement of the infinite-wavelength limit of the struc- ture factor,

    Ruobing Xie, Gabrielle G Long, Steven J Weigand, Si- mon C Moss, Tobi Carvalho, Sjoerd Roorda, Miroslav Hejna, Salvatore Torquato, and Paul J Steinhardt, “Hy- peruniformity in amorphous silicon based on the mea- surement of the infinite-wavelength limit of the struc- ture fact...

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    Uni- versal hidden order in amorphous cellular geometries,

    Michael A Klatt, Jakov Lovri´ c, Duyu Chen, Se- bastian C Kapfer, Fabian M Schaller, Philipp W A Sch¨ onh¨ ofer, Bruce S Gardiner, Ana-Sunˇ cana Smith, Gerd E Schr¨ oder-Turk, and Salvatore Torquato, “Uni- versal hidden order in amorphous cellular geometries,” Nature communica...

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    Quantum jamming transition to a correlated electron glass in 1t-tas 2,

    Y. A. Gerasimenko, I. Vaskivskyi, M. Litskevich, J. Ravnik, J. Vodeb, M. Diego, V. Kabanov, and D. Mi- hailovic, “Quantum jamming transition to a correlated electron glass in 1t-tas 2,” Nat. Mater. 18, 1078–1083 (2019)

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    Quan- tum phase transition between hyperuniform density dis- tributions,

    Shiro Sakai, Ryotaro Arita, and Tomi Ohtsuki, “Quan- tum phase transition between hyperuniform density dis- tributions,” arXiv preprint arXiv:2207.09698 (2022)

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    Disordered hyperuniform quasi-1d ma- terials,

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    Multihyperuniform long-range order in medium-entropy alloys,

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    Avian photoreceptor patterns represent a disordered hyperuniform solution to a multiscale pack- ing problem,

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    How a well-adapted im- mune system is organized,

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    Zhenpeng Ge, “The hidden order of turing patterns in arid and semi-arid vegetation ecosystems,” Proceedings of the National Academy of Sciences 120, e2306514120 (2023)

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    Universal hyperuniform organization in looped leaf vein networks,

    Yuan Liu, Duyu Chen, Jianxiang Tian, Wenxiang Xu, and Yang Jiao, “Universal hyperuniform organization in looped leaf vein networks,” Physical Review Letters 133, 028401 (2024)

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    Effect of imperfections on the hyperuniformity of many-body systems,

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    Local number fluctuations in hyperuniform and non- hyperuniform systems: Higher-order moments and dis- tribution functions,

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    Flow in random porous media: mathematical formulation, variational principles, and rigorous bounds,

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    1989

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    Generation and structural characterization of debye random media,

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    Understand- ing degeneracy of two-point correlation functions via debye random media,

    Murray Skolnick and Salvatore Torquato, “Understand- ing degeneracy of two-point correlation functions via debye random media,” Physical Review E 104, 045306 (2021)

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    Fluid permeabilities of triply periodic minimal surfaces,

    Y Jung and S Torquato, “Fluid permeabilities of triply periodic minimal surfaces,” Physical Review E 72, 056319 (2005)

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    Microstructural and transport characteristics of triply periodic bicontin- uous materials,

    Salvatore Torquato and Jaeuk Kim, “Microstructural and transport characteristics of triply periodic bicontin- uous materials,” Acta Materialia 276, 120142 (2024)

    2024

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.