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arXiv 2023年

作者： Kargaran, Somayeh Jüttler, Bert Takacs, Thomas Doctoral Program Computational Mathematics Johannes Kepler University Linz Altenberger Straße 69 Linz4040 Austria Hagenberg Austria Institute of Applied Geometry Johannes Kepler University Linz Altenberger Straße 69 Linz4040 Austria Austrian Academy of Sciences Altenberger Straße 69 Linz4040 Austria

Isogeometric analysis (IGA) is a numerical method, proposed in [1], that connects computer-aided design (CAD) with finite element analysis (FEA). In CAD the computational domain is usually represented by B-spline or NURBS patches. Given a B-spline or NURBS parameterization of the domain, an isogeometric discretization is defined on the domain using the same B-spline or NURBS basis as for the domain parameterization. Ideally, such an isogeometric discretization allows an exact representation of the underlying CAD model. CAD models usually represent only the boundary of the object. For planar domains, the CAD model is given as a collection of curves representing the boundary. Finding a suitable parameterization of the interior is one of the major issues for IGA, similar to the mesh generation process in the FEA setting. The objective of this isogeometric parameterization problem is to obtain a set of patches, which exactly represent the boundary of the domain and which are parameterized regularly and without self-intersections. This can be achieved by segmenting the domain into patches which are matching along interfaces, or by covering the domain with overlapping patches. In this paper we follow the second approach. To construct from a given boundary curve a planar parameterization suitable for IGA, we propose an offset-based domain parameterization algorithm. Given a boundary curve, we obtain an inner curve by generalized offsetting. The inner curve, together with the boundary curve, naturally defines a ring-shaped patch with an associated parameterization. By definition, the ring-shaped patch has a hole, which can be covered by a multi-cell domain. Consequently, the domain is represented as a union of two overlapping subdomains which are regularly parameterized. On such a configuration, one can employ the overlapping multi-patch (OMP) method, as introduced in [2], to solve PDEs on the given domain. The performance of the proposed method is reported in several numer

关键词： Parameterization

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Quantum phase transitions in long-range interacting hyperuniform spin chains in a transverse field

arXiv

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arXiv 2020年

作者： Bose, Amartya Torquato, Salvatore Department of Chemistry Princeton University PrincetonNJ08544 United States Department of Physics Princeton University PrincetonNJ08544 United States Princeton Institute for the Science and Technology of Materials Princeton University PrincetonNJ08544 United States Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

Hyperuniform states of matter are characterized by anomalous suppression of long-wavelength density fluctuations. While most of the interesting cases of disordered hyperuniformity are provided by complex many-body systems like liquids or amorphous solids, classical spin chains with certain long-range interactions have been shown to demonstrate the same phenomenon. Such systems involving spin chains are ideal models for exploring the effects of quantum mechanics on hyperuniformity. It is well-known that the transverse field Ising model shows a quantum phase transition (QPT) at zero temperature. Under the quantum effects of a transverse magnetic field, classical hyperuniform spin chains are expected to lose their hyperuniformity. High-precision simulations of these cases are complicated because of the presence of highly nontrivial long-range interactions. We perform extensive analysis of these systems using density matrix renormalization group (DMRG) to study the possibilities of phase transitions and the mechanism by which they lose hyperuniformity. Even for a spin chain of length 30, we see discontinuous changes in properties like the "τ order metric" of the ground state, the measure of hyperuniformity and the second cumulant of the total magnetization along the x-direction, all suggestive of first-order QPTs. An interesting feature of the phase transitions in these disordered hyperuniform spin chains is that, depending on the parameter values, the presence of transverse magnetic field may remarkably lead to increase in the order of the ground state as measured by the "τ order metric," even if hyperuniformity is lost. Therefore, it would be possible to design materials to target specific novel quantum behaviors in the presence of a transverse magnetic field. Our numerical investigations suggest that these spin chains can show no more than two QPTs. We further analyze the long-range interacting spin chains via the Jordan-Wigner mapping on to a system of spinless ferm

关键词： Phase transitions

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Liquid-Liquid Transition in Water from First Principles

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Physical Review Letters 2022年第25期129卷 255702-255702页

作者： Thomas E. Gartner, III Pablo M. Piaggi Roberto Car Athanassios Z. Panagiotopoulos Pablo G. Debenedetti Department of Chemistry Princeton University Princeton New Jersey 08544 USA Department of Physics Princeton University Princeton New Jersey 08544 USA Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA Princeton Institute for the Science and Technology of Materials Princeton University Princeton New Jersey 08544 USA Department of Chemical and Biological Engineering Princeton University Princeton New Jersey 08544 USA

A long-standing question in water research is the possibility that supercooled liquid water can undergo a liquid-liquid phase transition (LLT) into high- and low-density liquids. We used several complementary molecular simulation techniques to evaluate the possibility of an LLT in an ab initio neural network model of water trained on density functional theory calculations with the SCAN exchange correlation functional. We conclusively show the existence of a first-order LLT and an associated critical point in the SCAN description of water, representing the first definitive computational evidence for an LLT in water from first principles.

关键词： Classical statistical mechanics First order phase transitions First-principles calculations Liquid-liquid phase transition Phase transitions Thermodynamics Deep learning Density functional theory Molecular dynamics

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Realizability of Iso-g2 Processes via Effective Pair Interactions

arXiv

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arXiv 2022年

作者： Wang, Haina Stillinger, Frank H. Torquato, Salvatore Department of Chemistry Princeton University PrincetonNJ08544 United States Department of Chemistry Department of Physics Princeton Institute of Materials and Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States School of Natural Sciences Institute for Advanced Study 1 Einstein Drive PrincetonNJ08540 United States

An outstanding problem in statistical mechanics is the determination of whether prescribed functional forms of the pair correlation function g2(r) [or equivalently, structure factor S(k)] at some number density ρ can be achieved by d-dimensional many-body systems. The Zhang-Torquato conjecture states that any realizable set of pair statistics, whether from a nonequilibrium or equilibrium system, can be achieved by equilibrium systems involving up to two-body interactions. In light of this conjecture, we study the realizability problem of the nonequilibrium iso-g2 process, i.e., the determination of density-dependent effective potentials that yield equilibrium states in which g2 remains invariant for a positive range of densities. Using a precise inverse methodology that determines effective potentials that match hypothesized functional forms of g2(r) for all r and S(k) for all k, we show that the unit-step function g2, which is the zero-density limit of the hard-sphere potential, is remarkably numerically realizable up to the packing fraction = 0.49 for d = 1. For d = 2 and 3, it is realizable up to the maximum "terminal" packing fraction c = 1/2d, at which the systems are hyperuniform, implying that the explicitly known necessary conditions for realizability are sufficient up through c. For near but below c, the large-r behaviors of the effective potentials are given exactly by the functional forms exp[−κ()r] for d = 1, r−1/2 exp[−κ()r] for d = 2, and r−1 exp[−κ()r] (Yukawa form) for d = 3, where κ−1() is a screening length, and for = c, the potentials at large r are given by the pure Coulomb forms in the respective dimensions, as predicted by Torquato and Stillinger [Phys. Rev. E, 68, 041113 1-25 (2003)]. We also find that the effective potential for the pair statistics of the 3D "ghost" random sequential addition at the saturation density c = 1/8 is much shorter-ranged than that for the 3D unit-step-function g2 at c, and thus does not constrain the realizability

关键词： Inverse problems

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New tessellation-based procedure to design perfectly hyperuniform disordered dispersions for materials discovery

arXiv

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arXiv 2019年

作者： Kim, J. Torquato, S. Department of Physics Princeton University PrincetonNJ08544 United States Department of Chemistry Princeton University PrincetonNJ08544 United States Princeton Institute for the Science and Technology of Materials Princeton University PrincetonNJ08544 United States Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

Disordered hyperuniform dispersions are exotic amorphous two-phase materials characterized by an anomalous suppression of long-wavelength volume-fraction fluctuations, which endows them with novel physical properties. While such unusual materials have received considerable attention, a stumbling block has been an inability to create large samples that are truly hyperuniform due to current computational and experimental limitations. To overcome such barriers, we introduce a new and simple construction procedure that guarantees perfect hyperuniformity for very large sample sizes. This methodology involves tessellating space into cells and then inserting a particle into each cell such that the local-cell particle packing fractions are identical to the global packing fraction. We analytically prove that such dispersions are perfectly hyperuniform in the infinite-sample-size limit. Our methodology enables a remarkable mapping that converts a very large nonhyperuniform disordered dispersion into a perfectly hyperuniform one, which we numerically demonstrate in two and three dimensions. A similar analysis also establishes the hyperuniformity of the famous Hashin-Shtrikman multiscale dispersions, which possess optimal transport and elastic properties. Our hyperuniform designs can be readily fabricated using modern photolithographic and 3D printing technologies. The exploration of the enormous class of hyperuniform dispersions that can be designed and tuned by our tessellation-based methodology paves the way for accelerating the discovery of novel hyperuniform materials. Copyright © 2019, The Authors. All rights reserved.

关键词： 3D printers

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

arXiv

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arXiv 2021年

作者： Maher, Charles Emmett Stillinger, Frank H. Torquato, Salvatore Department of Chemistry Princeton University PrincetonNJ08544 United States Department of Physics Princeton University PrincetonNJ08544 United States Princeton Institute for the Science and Technology of Materials Princeton University PrincetonNJ08544 United States Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

Dense, disordered packings of particles are useful models of low-temperature amorphous phases of matter, biological systems, granular media, and colloidal systems. The study of dense packings of nonspherical particles enables one to ascertain how rotational degrees of freedom affect packing behavior. Here, we study superballs, a large family of deformations of the sphere, defined in three dimensions by |x1|2p + |x2|2p + |x3|2p ≤ 1, where p ∈ (0, ∞) is a deformation parameter indicating to what extent the shape deviates from a sphere. As p increases from the sphere point (p = 1), the superball tends to a cuboidal shape and approaches a cube in the p → ∞ limit. As p → 0.5, it approaches an octahedron, becomes a concave body with octahedral symmetry for p ¯) vary nonanalytically as p diverges from unity. Here, we use an event-driven molecular dynamics algorithm to produce MRJ superball packings with 0.85 ≤ p ≤ 1.50. To supplement the previous work on such packings, we characterize their large-scale structure by examining the behaviors of their structure factors S(Q) and spectral densities χV (Q), as the wave number Q tends to zero, and find that these packings are effectively hyperuniform for all values of p examined. We show that the mean width w¯ is a useful length scale to make distances dimensionless in order to compare systematically superballs of different shape. Moreover, we compute the complementary cumulative pore-size distribution F (δ) and find that the pore sizes tend to decrease as |1 − p| increases. From F (δ), we estimate how the fluid permeability, mean survival time, and principal diffusion relaxation time vary as a function of p. Additionally, we compute the diffusion "spreadability" S(t) [Torquato, Phys. Rev. E, 104, 054102, (2021)] of these packings and find that the long-time power-law scaling indicates these packings are hyperuniform with a small-Q power law scaling of the spectral density χV (Q) ∼ Qα with an exponent α that ranges from 0.64 at th

关键词： Spheres

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Designer Pair Statistics of Disordered Many-Particle Systems with Novel Properties

arXiv

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arXiv 2023年

作者： Wang, Haina Torquato, Salvatore Department of Chemistry Princeton University PrincetonNJ08544 United States Department of Chemistry Department of Physics Princeton Materials Institute Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States School of Natural Sciences Institute for Advanced Study 1 Einstein Drive PrincetonNJ08540 United States

Knowledge of exact analytical functional forms for the pair correlation function g2(r) and its corresponding structure factor S(k) of disordered many-particle systems is limited. For fundamental and practical reasons, it is highly desirable to add to the existing data base of analytical functional forms for such pair statistics. Here, we design a plethora of such pair functions in direct and Fourier spaces across the first three Euclidean space dimensions that are realizable by diverse many-particle systems with varying degrees of correlated disorder across length scales, spanning a wide spectrum of hyperuniform, typical nonhyperuniform and antihyperuniform ones. This is accomplished by utilizing an efficient inverse algorithm that determines equilibrium states with up to pair interactions at positive temperature that precisely match targeted forms for both g2(r) and S(k). Among other results, we realize an example with the strongest hyperuniform property among known positive-temperature equilibrium states, critical-point systems (implying unusual 1D systems with phase transitions) that are not in the Ising universality class, systems that attain self-similar pair statistics under Fourier transformation, and an experimentally feasible polymer model. We show that our pair functions enable one to achieve systems with a wide range of translational order and self-diffusion coefficients D, which are inversely related to one another. One can design other realizable pair statistics via linear combinations of our functions or by applying our inverse procedure to other desirable functional forms. Our approach facilitates the inverse design of materials with desirable physical and chemical properties by tuning their pair statistics. © 2023, CC BY.

关键词： Statistics

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Local Order Metrics for Two-Phase Media Across Length Scales

arXiv

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arXiv 2022年

作者： Torquato, Salvatore Skolnick, Murray Kim, Jaeuk Department of Chemistry Department of Physics Princeton Institute for the Science and Technology of Materials Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Department of Chemistry Princeton University PrincetonNJ08544 United States McKetta Department of Chemical Engineering University of Texas at Austin AustinTX78712 United States

The capacity to devise order metrics for microstructures of multiphase heterogeneous media is a highly challenging task, given the richness of the possible geometries and topologies of the phases that can arise. This investigation initiates a program to formulate order metrics to characterize the degree of order/disorder of the microstructures of two-phase media in d-dimensional Euclidean space Rd across length scales. In particular, we propose the use of the local volume-fraction variance σV2 (R) associated with a spherical window of radius R as an order metric. We determine σV2 (R) as a function of R for 22 different models across the first three space dimensions, including both hyperuniform and nonhyperuniform systems with varying degrees of short- and long-range order. We find that the local volume-fraction variance as well as asymptotic coefficients and integral measures derived from it provide reasonably robust and sensitive order metrics to categorize disordered and ordered two-phase media across all length scales. © 2022, CC BY-SA.

关键词： Volume fraction

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Nonlocal effective electromagnetic wave characteristics of composite media: Beyond the quasistatic regime

arXiv

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arXiv 2020年

作者： Torquato, Salvatore Kim, Jaeuk Department of Physics Princeton University PrincetonNJ08544 United States Department of Chemistry Princeton University PrincetonNJ08544 United States Princeton Institute Science and Technology of Materials Princeton University PrincetonNJ08544 United States Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

We derive exact nonlocal homogenized constitutive relations for the effective electromagnetic wave properties of disordered two-phase composites and metamaterials from first principles. This exact formalism enables us to extend the long-wavelength limitations of conventional homogenization estimates of the effective dynamic dielectric constant tensor Εe(kI, ω) for arbitrary microstructures so that it can capture spatial dispersion well beyond the quasistatic regime (where ω and kI are the frequency and wave vector of the incident radiation). We accomplish this task by deriving nonlocal strong-contrast expansions that exactly account for complete microstructural information (infinite set of n-point correlation functions) and hence multiple scattering to all orders for the range of wave numbers for which our extended homogenization theory applies, i.e., 0 ≤ |kI|. 1 (where is a characteristic heterogeneity length scale). Because of the fast-convergence properties of such expansions, their lower-order truncations yield accurate closed-form approximate formulas for Εe(kI, ω) that apply for a wide class of microstructures. These nonlocal formulas are resummed representations of the strong-contrast expansions that still accurately capture multiple scattering to all orders via the microstructural information embodied in the spectral density, which is easy to compute for any composite. The accuracy of these microstructure-dependent approximations is validated by comparison to full-waveform simulation computations for both 2D and 3D ordered and disordered models of composite media. Thus, our closed-form formulas enable one to predict accurately and efficiently the effective wave characteristics well beyond the quasistatic regime for a wide class of composite microstructures without having to perform full-blown simulations. We find that disordered hyperuniform media are generally less lossy than their nonhyperuniform counterparts. We also show that certain disordered hyperunif

关键词： Microstructure

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Random scalar fields and hyperuniformity

arXiv

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arXiv 2017年

作者： Ma, Zheng Torquato, Salvatore Department of Physics Princeton University PrincetonNJ08544 United States Department of Chemistry Princeton University PrincetonNJ08544 United States Princeton Institute for the Science and Technology of Materials Princeton University PrincetonNJ08544 United States Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

Disordered many-particle hyperuniform systems are exotic amorphous states of matter that lie between crystals and liquids. Hyperuniform systems have attracted recent attention because they are endowed with novel transport and optical properties. Recently, the hyperuniformity concept has been generalized to characterize two-phase media, scalar fields and random vector fields. In this paper, we devise methods to explicitly construct hyperuniform scalar fields. Specifically, we analyze spatial patterns generated from Gaussian random fields, which have been used to model the microwave background radiation and heterogeneous materials, the Cahn-Hilliard equation for spinodal decomposition, and Swift-Hohenberg equations that have been used to model emergent pattern formation, including Rayleigh-Bénard convection. We show that the Gaussian random scalar fields can be constructed to be hyperuniform. We also numerically study the time evolution of spinodal decomposition patterns and demonstrate that they are hyperuniform in the scaling regime. Moreover, we find that labyrinth-like patterns generated by the Swift-Hohenberg equation are effectively hyperuniform. We show that thresholding (level-cutting) a hyperuniform Gaussian random field to produce a two-phase random medium tends to destroy the hyperuniformity of the progenitor scalar field. We then propose guidelines to achieve effectively hyperuniform two-phase media derived from thresholded non-Gaussian fields. Our investigation paves the way for new research directions to characterize the large-structure spatial patterns that arise in physics, chemistry, biology and ecology. Moreover, our theoretical results are expected to guide experimentalists to synthesize new classes of hyperuniform materials with novel physical properties via coarsening processes and using state-of-the-art techniques, such as stereolithography and 3D printing. Copyright © 2017, The Authors. All rights reserved.

关键词： Gaussian distribution

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