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Correlations in interacting electron liquids: Many-body statistics and hyperuniformity

Physical Review B 2024年第10期110卷 104201页

作者： Haina Wang Rhine Samajdar Salvatore Torquato Department of Chemistry Department of Physics Princeton Center for Theoretical Science Princeton Materials Institute Program in Applied and Computational Mathematics

Disordered hyperuniform many-body systems are exotic states of matter with novel optical, transport, and mechanical properties. These systems are characterized by an anomalous suppression of large-scale density fluctuations compared to ordinary liquids. The structure factor of disordered hyperuniform systems often obeys the scaling relation S(k)∼Bkα with B,α>0 in the limit k→0. Ground states of d-dimensional free fermionic gases, which are fundamental models for many metals and semiconductors, are key examples of quantum disordered hyperuniform states with important connections to random matrix theory. However, the effects of electron-electron interactions as well as the polarization of the electron liquid on hyperuniformity have not been explored thus far. In this paper, we systematically address these questions by deriving the analytical small-k behaviors (and, associated, α and B) of the total and spin-resolved structure factors of quasi-one-dimensional, two-dimensional, and three-dimensional electron liquids for varying polarizations and interaction parameters. We validate that these equilibrium disordered ground states are hyperuniform, as dictated by the fluctuation-compressibility relation. Interestingly, free fermions, partially polarized interacting fermions, and fully polarized interacting fermions are characterized by different values of the small-k scaling exponent α and coefficient B. In particular, partially polarized fermionic liquids exhibit a unique form of multihyperuniformity, in which the net configuration exhibits a stronger form of hyperuniformity (i.e., larger α) than each individual spin component. The detailed theoretical analysis of such small-k behaviors enables the construction of corresponding equilibrium classical systems under effective one- and two-body interactions that mimic the pair statistics of quantum electron liquids. Our paper thus reveals that highly unusual hyperuniform and multihyperuniform states can be achieved in simple

关键词： Electronic structure Quantum many-body systems Two-dimensional electron gas

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Positive- and negative-effective-viscosity phenomena in isotropic and anisotropic Beltrami flows

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Physical Review A 1986年第1期34卷 381-381页

作者： B. J. Bayly V. Yakhot Program in Applied and Computational Mathematics Princeton University Fine Hall Princeton New Jersey 08544

A field-theoretic approach, analogous to Kraichnan’s direct-interaction approximation, to the stability theory of complex three-dimensional flows is developed. The long-wavelength stability of a class of Beltrami flows in an unbounded, viscous fluid is considered. We examine two flows in detail, to illustrate the effects of strong isotropy versus strong anisotropy in the basic flow. The effect of the small-scale flow on the long-wavelength perturbations may be interpreted as an effective viscosity. Using diagrammatic techniques, we construct the first-order smoothing and direct-interaction approximations for the perturbation dynamics. It is argued that the effective viscosity for the isotropic flow is always positive, and approaches a value independent of the molecular viscosity in the high-Reynolds-number limit; this flow is thus stable to long-wavelength disturbances. The anisotropic flow has negative effective viscosity for some orientations of the disturbance, and is therefore unstable, when its Reynolds number exceeds √2 .

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The Weyl-Heisenberg ensemble: Statistical mechanics meets time-frequency analysis

The Weyl-Heisenberg ensemble: Statistical mechanics meets ti...

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International Conference on Sampling Theory and Applications

作者： Luis Daniel Abreu Joao Pereira Jose Luis Romero Salvatore Torquato Austrian Academy of Sciences Acoustics Research Institute Program in Applied and Computational Mathematics Princeton University Department of Chemistry Princeton Materials Institute and Program in Applied and Computational Mathematics Princeton University

ISBN: (纸本)9781538615669

We present our recent work on the Weyl-Heisenberg ensemble and its statistical properties [4]. The WH ensemble is a class of determinantal point processes associated with the Schrodinger representation of the Heisenberg group. As a special example, WH ensembles include a multi-layer extension of the Ginibre ensemble modeling the distribution of electrons in higher Landau levels. We describe the hyperuniformity of WH ensembles, which characterizes a state of matter for which (scaled) density fluctuations diminish towards zero at the largest length scales. Our approach is based on methods from time-frequency analysis. We introduce the main results from [4] highlighting time-frequency techniques and connections to the theory of polyanalytic functions, and also present some small extensions.

关键词： Weyl-Heisenberg ensemble Hyperuniformity Higher Landau levels Disorder Fluctuations

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Landauʼs necessary density conditions for the Hankel transform

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Journal of Functional Analysis 2012年第4期262卷 1845-1866页

作者： Luís Daniel Abreu Afonso S. Bandeira Department of Mathematics of University of Coimbra 3001-454 Coimbra Portugal Program in Applied and Computational Mathematics Princeton University NJ 08544 USA

We will prove an analogue of Landauʼs necessary conditions [H.J. Landau, Necessary density conditions for sampling and interpolation of certain entire functions, Acta Math. 117 (1967) 37–52] for spaces of functions whose Hankel transform is supported in a measurable subset S of the positive semi-axis. As a special case, necessary density conditions for the existence of Fourier–Bessel frames are obtained.

关键词： Sampling and interpolation Beurling–Landau density Hankel transform Bessel functions Fourier–Bessel frames

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Novel Conservative Methods for Schrödinger Equations with Variable Coefficients over Long Time

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Communications in computational Physics 2014年第3期15卷 692-711页

作者： Xu Qian Yaming Chen Songhe Song Department of Mathematics and Systems Science and State Key Laboratory of High Performance ComputingNational University of Defense TechnologyChangsha 410073P.R.China Program in Applied and Computational Mathematics Princeton UniversityPrincetonNJ 08544USA School of Mathematical Sciences Queen Mary University of LondonLondon E14NSUK

In this paper,we propose a wavelet collocation splitting(WCS)method,and a Fourier pseudospectral splitting(FPSS)method as comparison,for solving onedimensional and two-dimensional Schrödinger equations with variable coefficients in quantum *** two methods can preserve the intrinsic properties of original problems as much as *** splitting technique increases the computational ***,the error estimation and some conservative properties are *** is proved to preserve the charge conservation *** global energy and momentum conservation laws can be preserved under several *** experiments are conducted during long time computations to show the performances of the proposed methods and verify the theoretical analysis.

关键词： Schrödinger equation wavelet collocation method splitting technique conservative property

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Deep potentials for materials science

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Materials Futures 2022年第2期1卷 89-115页

作者： Tongqi Wen Linfeng Zhang Han Wang Weinan E David J Srolovitz Department of Mechanical Engineering The University of Hong KongHong KongHong Kong Special Administrative Region of China DP Technology BeijingPeople’s Republic of China AI for Science Institute BeijingPeople’s Republic of China Laboratory of Computational Physics Institute of Applied Physics and Computational MathematicsBeijingPeople’s Republic of China HEDPS CAPTCollege of EngineeringPeking UniversityBeijingPeople’s Republic of China School of Mathematical Sciences Peking UniversityBeijingPeople’s Republic of China Department of Mathematics and Program in Applied and Computational Mathematics Princeton UniversityPrincetonNJUnited States of America International Digital Economy Academy(IDEA) ShenzhenPeople’s Republic of China

To fill the gap between accurate(and expensive)ab initio calculations and efficient atomistic simulations based on empirical interatomic potentials,a new class of descriptions of atomic interactions has emerged and been widely applied;*** learning potentials(MLPs).One recently developed type of MLP is the deep potential(DP)*** this review,we provide an introduction to DP methods in computational materials *** theory underlying the DP method is presented along with a step-by-step introduction to their development and *** also review materials applications of DPs in a wide range of materials *** DP Library provides a platform for the development of DPs and a database of extant *** discuss the accuracy and efficiency of DPs compared with ab initio methods and empirical potentials.

关键词： deep potential atomistic simulation machine learning potential neural network

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Diffusion maps - A probabilistic interpretation for spectral embedding and clustering algorithms

Diffusion maps - A probabilistic interpretation for spectral...

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作者： Nadler, Boaz Lafon, Stephane Coifman, Ronald Kevrekidis, Ioannis G. Department of Computer Science and Applied Mathematics Weizmann Institute of Science Rehovot 76100 Israel Google Inc. United States Department of Mathematics Yale University New Haven CT 06520-8283 United States Department of Chemical Engineering Program in Applied and Computational Mathematics Princeton University Princeton NJ 08544 United States

ISBN: (纸本)9783540737490

Spectral embedding and spectral clustering are common methods for non-linear dimensionality reduction and clustering of complex high dimensional datasets. In this paper we provide a diffusion based probabilistic analysis of algorithms that use the normalized graph Laplacian. Given the pairwise adjacency matrix of all points in a dataset, we define a random walk on the graph of points and a diffusion distance between any two points. We show that the diffusion distance is equal to the Euclidean distance in the embedded space with all eigenvectors of the normalized graph Laplacian. This identity shows that characteristic relaxation times and processes of the random walk on the graph are the key concept that governs the properties of these spectral clustering and spectral embedding algorithms. Specifically, for spectral clustering to succeed, a necessary condition is that the mean exit times from each cluster need to be significantly larger than the largest (slowest) of all relaxation times inside all of the individual clusters. For complex, multiscale data, this condition may not hold and multiscale methods need to be developed to handle such situations.

关键词： Relaxation time

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Scalable Cluster-Consistency Statistics for Robust Multi-Object Matching

Scalable Cluster-Consistency Statistics for Robust Multi-Obj...

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International Conference on 3D Imaging, Modeling, Processing, Visualization and Transmission (3DIMPVT)

作者： Yunpeng Shi Shaohan Li Tyler Maunu Gilad Lerman Program in Applied and Computational Mathematics Princeton University School of Mathematics University of Minnesota Brandeis University

ISBN: (纸本)9781665426893

We develop new statistics for robustly filtering corrupted keypoint matches in the structure from motion pipeline. The statistics are based on consistency constraints that arise within the clustered structure of the graph of keypoint matches. The statistics are designed to give smaller values to corrupted matches and than uncorrupted matches. These new statistics are combined with an iterative reweighting scheme to filter keypoints, which can then be fed into any standard structure from motion pipeline. This filtering method can be efficiently implemented and scaled to massive datasets as it only requires sparse matrix multiplication. We demonstrate the efficacy of this method on synthetic and real structure from motion datasets and show that it achieves state-of-the-art accuracy and speed in these tasks.

关键词： Matched filters Structure from motion Three-dimensional displays Filtering Image edge detection Pipelines FCC

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RECURSIVE SAMPLING OF PLANAR GRAPHS AND FRACTAL PROPERTIES OF A TWO-DIMENSIONAL QUANTUM GRAVITY

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International Journal of Modern Physics C 1990年第1期01卷 165-179页

作者： MICHAEL E. AGISHTEIN ALEXANDER A. MIGDAL Physics Department and Program in Applied and Computational Mathematics Princeton University Fine Hall Princeton NJ 08544 USA

We describe a new approach to the Monte-Carlo simulations of two-dimensional gravity. Standard dynamical triangulation technique was combined with results of direct enumeration of the cubic graphs. As a result we were able to build large (128K vertices) statistically independent random graphs directly. The quantitative correspondence between our results and those obtained by standard methods has been observed. The algorithm proved to be so efficient that we were able to conduct all the simulations, which usually require the most powerful computers, on an Iris workstation. An opportunity to generate large random graphs allowed us to observe that the internal geometry of random surfaces is more complicated than simple fractals. External geometry also proved to be rather peculiar.

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Computing Optimal Interfacial Structure of Modulated Phases

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Communications in computational Physics 2017年第1期21卷 1-15页

作者： Jie Xu Chu Wang An-Chang Shi Pingwen Zhang LMAM and School of Mathematical Sciences Peking UniversityBeijing 100871China Program in Applied and Computational Mathematics Princeton UniversityPrincetonNew Jersey 08544USA Department of Physics and Astronomy McMaster UniversityHamiltonOntario L8S4M1Canada

We propose a general framework of computing interfacial structures between two modulated *** we propose to use a computational box consisting of two half spaces,each occupied by a modulated phase with given position and *** boundary conditions and basis functions are chosen to be commensurate with the bulk *** observe that the ordered nature of modulated structures stabilizes the interface,which enables us to obtain optimal interfacial structures by searching local minima of the free energy *** framework is applied to the Landau-Brazovskii model to investigate interfaces between modulated phases with different relative positions and *** types of novel complex interfacial structures emerge from the calculations.

关键词： Interface modulated phase metastable state compatibility Landau-Brazovskiimodel

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