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arXiv 2020年

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

The local number variance σ2(R) associated with a spherical sampling window of radius R enables a classification of many-particle systems in d-dimensional Euclidean space Rd according to the degree to which large-scale density fluctuations are suppressed, resulting in a demarcation between hyperuniform and nonhyperuniform phyla. To more completely characterize density fluctuations, we carry out an extensive study of higher-order moments or cumulants, including the skewness γ1(R), excess kurtosis γ2(R) and the corresponding probability distribution function P[N(R)] of a large family of models across the first three space dimensions, including both hyperuniform and nonhyperuniform systems with varying degrees of short- and long-range order. To carry out this comprehensive program, we derive new theoretical results that apply to general point processes and conduct high-precision numerical studies. Specifically, we derive explicit closed-form integral expressions for γ1(R) and γ2(R) that encode structural information up to three-body and four-body correlation functions, respectively. We also derive rigorous bounds on γ1(R), γ2(R) and P[N(R)] for general point processes and corresponding exact results for general packings of identical spheres. High-quality simulation data for γ1(R), γ2(R) and P[N(R)] are generated for each model. We also ascertain the proximity of P[N(R)] to the normal distribution via a novel Gaussian "distance" metric l2(R). Among all models, the convergence to a central limit theorem (CLT) is generally fastest for the disordered hyperuniform processes such that γ1(R) ∼ l2(R) ∼ R-(d+1)/2 and γ2(R) ∼ R-(d+1) for large R. The convergence to a CLT is slower for standard nonhyperuniform models and slowest for the "antihyperuniform" model studied here. We prove that one-dimensional hyperuniform systems of class I or any d-dimensional lattice cannot obey a CLT. Remarkably, we discovered that the gamma distribution provides a good approximation to P[N(R)] for

关键词： Higher order statistics

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Cloaking the underlying long-range order of randomly perturbed lattices

arXiv

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arXiv 2020年

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

Random, uncorrelated displacements of particles on a lattice preserve the hyperuniformity of the original lattice, that is, normalized density fluctuations vanish in the limit of infinite wavelengths. In addition to a diffuse contribution, the scattering intensity from the the resulting point pattern typically inherits the Bragg peaks (long-range order) of the original lattice. Here we demonstrate how these Bragg peaks can be hidden in the effective diffraction pattern of independent and identically distributed perturbations. All Bragg peaks vanish if and only if the sum of all probability densities of the positions of the shifted lattice points is a constant at all positions. The underlying long-range order is then ‘cloaked’ in the sense that it cannot be reconstructed from the pair correlation function alone. On the one hand, density fluctuations increase monotonically with the strength of perturbations a, as measured by the hyperuniformity order metric Λ. On the other hand, the disappearance and reemergence of long-range order, depending on whether the system is cloaked or not as the perturbation strength increases, is manifestly captured by the τ order metric. Therefore, while the perturbation strength a may seem to be a natural choice for an order metric of perturbed lattices, the τ order metric is a superior choice. It is noteworthy that cloaked perturbed lattices allow one to easily simulate very large samples (with at least 106 particles) of disordered hyperuniform point patterns without Bragg peaks. Copyright © 2020, The Authors. All rights reserved.

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Difficulty in predicting shallow defects with hybrid functionals: Implication of the long-range exchange interaction

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Physical Review B 2013年第3期88卷 035134-035134页

作者： Junhyeok Bang Y. Y. Sun Tesfaye A. Abtew Amit Samanta Peihong Zhang S. B. Zhang Department of Physics Applied Physics and Astronomy Rensselaer Polytechnic Institute Troy New York 12180 USA Department of Physics University at Buffalo State University of New York Buffalo New York 14260 USA Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08540 USA

We identify an important issue in defect studies using hybrid functionals. When modeling a defect, which is supposedly an isolated system, with a finite-size supercell, the inclusion of a fraction of the Hartree-Fock interaction results in a strong cell-size dependence and an extremely slow convergence of the calculated defect properties, especially for shallow defect. These behaviors may give rise to a number of errors in calculated defect properties, including the deepening of transition level and overstabilization of shallow defects. Numerical results from hybrid functional calculations for a diverse array of systems can be understood within the Hartree-Fock theory of an electron-gas model, indicating that the long-range exchange is the main cause for the errors in the calculated defect properties within hybrid functionals.

关键词： EXCHANGE interactions (Magnetism) HARTREE-Fock approximation ELECTRON gas MATHEMATICAL models FUNCTIONS (mathematics) ERRORS

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Modeling binary time series using gaussian processes with application to predicting sleep states

arXiv

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arXiv 2017年

作者： Gao, Xu Shahbaba, Babak Ombao, Hernando Department of Statistics University of California IrvineCA United States Department of Cognitive Sciences University of California IrvineCA United States Program on Applied Mathematics and Computational Science King Abdullah University of Science and Technology

Motivated by the problem of predicting sleep states, we develop a mixed effects model for binary time series with a stochastic component represented by a Gaussian process. The fixed component captures the effects of covariates on the binary-valued response. The Gaussian process captures the residual variations in the binary response that are not explained by covariates and past realizations. We develop a frequentist modeling framework that provides efficient inference and more accurate predictions. Results demonstrate the advantages of improved prediction rates over existing approaches such as logistic regression, generalized additive mixed model, models for ordinal data, gradient boosting, decision tree and random forest. Using our proposed model, we show that previous sleep state and heart rates are significant predictors for future sleep states. Simulation studies also show that our proposed method is promising and robust. To handle computational complexity, we utilize Laplace approximation, golden section search and successive parabolic interpolation. With this paper, we also submit an R-package (HIBITS) that implements the proposed procedure. Copyright © 2017, The Authors. All rights reserved.

关键词： Gaussian noise (electronic)

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Importance of reference Hamiltonians containing exact exchange for accurate one-shot GW calculations of Cu2O

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Physical Review B 2012年第23期85卷 235142-235142页

作者： Leah Y. Isseroff Emily A. Carter Department of Chemical and Biological Engineering Princeton University Princeton New Jersey 08544-5263 USA Department of Mechanical and Aerospace Engineering Program in Applied and Computational Mathematics and Andlinger Center for Energy and the Environment Princeton University Princeton New Jersey 08544-5263 USA

We show that a “one-shot” GW approach (denoted G0W0) can accurately calculate the photoemission/inverse-photoemission properties of Cu2O. As the results of any perturbative method are heavily dependent on the reference state, the appropriate reference Hamiltonian for G0W0 is identified by evaluating the performance of density-functional-theory-based input wave functions and eigenvalues generated with selected exchange-correlation functionals. It is shown that a reference Hamiltonian employing the hybrid Heyd-Scuseria-Ernzerhof functional used in conjunction with G0W0 produces an accurate photoemission/inverse-photoemission band gap and photoemission spectrum whose character is then further analyzed. The physical origin of why a hybrid functional is required for the zeroth-order wave function is discussed, giving insight into the unique electronic structure of Cu2O in comparison to other transition-metal oxides.

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Limited angle acousto-electrical tomography

arXiv

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arXiv 2017年

作者： Hubmer, Simon Knudsen, Kim Li, Changyou Sherina, Ekaterina Johannes Kepler University Linz Doctoral Program Computational Mathematics Altenbergerstraße 69 LinzA-4040 Austria Technical University of Denmark Department of Applied Mathematics and Computer Science Asmussens Allé Kongens Lyngby2800 Denmark Technical University of Denmark Department of Applied Mathematics and Computer Science Asmussens Allé Kongens Lyngby2800 Denmark School of Electronics and Information Northwestern Polytechnical University Xian China Technical University of Denmark Department of Applied Mathematics and Computer Science Asmussens Allé Kongens Lyngby2800 Denmark

This paper considers the reconstruction problem in Acousto-Electrical Tomography, i.e., the problem of estimating a spatially varying conductivity in a bounded domain from measurements of the internal power densities resulting from different prescribed boundary conditions. Particular emphasis is placed on the limited angle scenario, in which the boundary conditions are supported only on a part of the boundary. The reconstruction problem is formulated as an optimization problem in a Hilbert space setting and solved using Landweber iteration. The resulting algorithm is implemented numerically in two spatial dimensions and tested on simulated data. The results quantify the intuition that features close to the measurement boundary are stably reconstructed and features further away are less well reconstructed. Finally, the ill-posedness of the limited angle problem is quantified numerically using the singular value decomposition of the corresponding linearized problem.65J22, 35R30, 65M32 Copyright © 2017, The Authors. All rights reserved.

关键词： Boundary conditions

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Synthetic diamond and wurtzite structures self-assemble with isotropic pair interactions

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Physical Review E 2007年第3期75卷 031403-031403页

作者： Mikael C. Rechtsman Frank H. Stillinger Salvatore Torquato Department of Physics Princeton University Princeton New Jersey 08544 USA Department of Chemistry Princeton University Princeton New Jersey 08544 USA Program in Applied and Computational Mathematics and PRISM Princeton New Jersey 08544 USA Princeton Center for Theoretical Physics Princeton New Jersey 08544 USA

Using inverse statistical-mechanical optimization techniques, we have discovered isotropic pair interaction potentials with strongly repulsive cores that cause the tetrahedrally coordinated diamond and wurtzite lattices to stabilize, as evidenced by lattice sums, phonon spectra, positive-energy defects, and self-assembly in classical molecular dynamics simulations. These results challenge conventional thinking that such open lattices can only be created via directional covalent interactions observed in nature. Thus, our discovery adds to fundamental understanding of the nature of the solid state by showing that isotropic interactions enable the self-assembly of open crystal structures with a broader range of coordination number than previously thought. Our work is important technologically because of its direct relevance generally to the science of self-assembly and specifically to photonic crystal fabrication.

关键词： CRYSTALS NANOPARTICLES

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End-To-End Quantum Machine Learning Implemented with Controlled Quantum Dynamics

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Physical Review applied 2020年第6期14卷 064020-064020页

作者： Re-Bing Wu Xi Cao Pinchen Xie Yu-xi Liu Department of Automation Tsinghua University Beijing 100084 China Beijing National Research Center for Information Science and Technology Beijing 100084 China Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA Institute of Microelectronics Tsinghua University Beijing 100084 China

To achieve quantum machine learning on imperfect noisy intermediate-scale quantum (NISQ) processors, the entire physical implementation should include as few as possible hand-designed modules with only a few ad hoc parameters to be determined. This work presents such a hardware-friendly end-to-end quantum-machine-learning scheme that can be implemented with imperfect NISQ processors. The proposal transforms the machine-learning task to the optimization of controlled quantum dynamics, in which the learning model is parameterized by experimentally tunable control variables. Our design also enables automated feature selection by encoding the raw input to quantum states through agent control variables. Comparing with the gate-based parameterized quantum circuits, the proposed end-to-end quantum-learning model is easy to implement as there are only a few ad hoc parameters to be determined. Numerical simulations on the benchmarking MNIST (Mixed National Institute of Standards and Technology) dataset of handwritten digits demonstrate that the model can achieve high performance using only 3–5 qubits without downsizing the dataset, which shows great potential for accomplishing large-scale real-world learning tasks on NISQ processors.

关键词： Machine learning Noise Quantum control Quantum information architectures & platforms Artificial neural networks

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Edge-Stress-Induced Warping of Graphene Sheets and Nanoribbons

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Physical Review Letters 2008年第24期101卷 245501-245501页

作者： V. B. Shenoy C. D. Reddy A. Ramasubramaniam Y. W. Zhang Division of Engineering Brown University Providence Rhode Island 02912 USA Institute of High Performance Computing Singapore 138632 Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA Department of Materials Science and Engineering National University of Singapore Singapore 119260

We show that edge stresses introduce intrinsic ripples in freestanding graphene sheets even in the absence of any thermal effects. Compressive edge stresses along zigzag and armchair edges of the sheet cause out-of-plane warping to attain several degenerate mode shapes. Based on elastic plate theory, we identify scaling laws for the amplitude and penetration depth of edge ripples as a function of wavelength. We also demonstrate that edge stresses can lead to twisting and scrolling of nanoribbons as seen in experiments. Our results underscore the importance of accounting for edge stresses in thermal theories and electronic structure calculations for freestanding graphene sheets.

关键词： Nanoribbons

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Multifunctional Hyperuniform Cellular Networks: Optimality, Anisotropy and Disorder

arXiv

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arXiv 2018年

作者： Torquato, S. Chen, D. 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

Disordered hyperuniform heterogeneous materials are new, exotic amorphous states of matter that behave like crystals in the manner in which they suppress volume-fraction fluctuations at large length scales, and yet are statistically isotropic with no Bragg peaks. It has recently been shown that disordered hyperuniform dielectric two-dimensional cellular network solids possess complete photonic band gaps comparable in size to photonic crystals, while at the same time maintaining statistical isotropy, enabling waveguide geometries not possible with photonic crystals. Motivated by these developments, we explore other functionalities of various two-dimensional ordered and disordered hyperuniform cellular networks, including their effective thermal or electrical conductivities and elastic moduli. We establish the multifunctionality of a class of such low-density networks by demonstrating that they maximize or virtually maximize the effective conductivities and elastic moduli. This is accomplished using the machinery of homogenization theory, including optimal bounds and cross-property bounds, and statistical mechanics. We rigorously prove that anisotropic networks consisting of sets of intersecting parallel channels in the low-density limit, ordered or disordered, possess optimal effective conductivity tensors. For a variety of different disordered networks, we show that when short-range and long-range order increases, there is an increase in both the effective conductivity and elastic moduli of the network. Moreover, we demonstrate that the effective conductivity and elastic moduli of various disordered networks derived from disordered "stealthy" hyperuniform point patterns possess virtually optimal values. We note that the optimal networks for conductivity are also optimal for the fluid permeability associated with slow viscous flow through the channels as well as the mean survival time associated with diffusion-controlled reactions in the channels. In summary, we have

关键词： Elastic moduli

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