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A molecular dynamical study of granular fluids I: The unforced granular gas in two dimensions

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Journal of Scientific Computing 1993年第1期8卷 1-40页

作者： Goldhirsch, I. Tan, M.-L. Zanetti, G. Department of Fluid Mechanics and Heat Transfer Faculty of Engineering Tel-Aviv University Ramat Aviv 69978 Tel-Aviv Israel Program in Applied and Computational Mathematics Princeton University Princeton 08544 New Jersey United States

Results of a numerical study of the dynamics of a collection of disks colliding inelastically in a periodic two-dimensional enclosure are presented. The properties of this system, which is perhaps the simplest model for rapidly flowing granular materials, are markedly different from those known for atomic or moleclar gases, in which collisions are of elastic nature. The most prominent feature characterizing granular systems, even in the idealized situation in which no external forcing exists and the initial condition is statistically homogeneous, is their inherent instability to inhomogeneous fluctuations. Granular "gases" are thus generically nonuniform, a fact that suggests extreme caution in pursuing direct analogies with molecular gases. We find that once an inhomogeneous state sets in, the velocity distribution functions differ from the classical Maxwell-Boltzmann distribution. Other characteristics of the system are different from their counterparts in molecular systems as well. For a given value of the coefficient of restitution, e, a granular system forms clusters of typical separation L₀≈l/(1-e²)^1/2, where l is the mean free path in the corresponding homogeneous system. Most of the fluctuating kinetic energy then resides in the relatively dilute regions that surround the clusters. Systems whose linear dimensions are less then L⁰ do not give rise to clusters;still they are inhomogeneous, the scale of the corresponding inhomogeneity being the longest wavelength allowed by the system's size. The present article is devoted to a detailed numerical study of the above-mentioned clustering phenomenon in two dimensions and in the absence of external forcing. A theoretical framework explaining this phenomenon is outlined. Some general implications as well as practical ramifications are discussed. © 1993 Plenum Publishing Corporation.

关键词： Granular flow dynamics molecular dynamics numerical simulation

Three-dimensional dynamics and transition to turbulence in the wake of bluff objects

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Journal of Fluid Mechanics 1992年第1期238卷 1-30页

作者： Karniadakis, George Em Triantafyllou, George S. Department of Mechanical and Aerospace Engineering and Program of Applied and Computational Mathematics Princeton University Princeton NJ 08544 United States The Levich Institute and Department of Mechanical Engineering The City College of New York New York NY 10031 United States

The wakes of bluff objects and in particular of circular cylinders are known to undergo a ‘fast ’ transition, from a laminar two-dimensional state a t Reynolds number 200 to a turbulent state a t Reynolds number 400. The process has been documented in several eXperimental mvestigations, but the underlying physical mechanisms have remained largely unknown so far. In this paper, the transition process is investigated numerically, through direct simulation of the NavierStokes equations at representative Reynolds numbers, up to 500. A high-order timeaccurate, miXed spectral/spectral element technique is used. It is shown that the wake first becomes three-dimensional, as a result of a secondary instability of the two-dimensional vorteX street. This secondary instability appears at a Reynolds number close to 200. For slightly supercritical Reynolds numbers, a harmonic state develops, in which the flow oscillates at its fundamental frequency (Strouhal number) around a spanwise modulated time-average flow. In the near wake the modulation wavelength of the time-average flow is half of the spanwise wavelength of the perturbation flow, consistently with linear instability theory. The vorteX filaments have a spanwise wavy shape in the near wake, and form rib-like structures further downstream. At higher Reynolds numbers the three-dimensional flow oscillation undergoes a period-doubling bifurcation, in which the flow alternates between two different states. Phase-space analysis of the flow shows that the basic limit cycle has branched into two connected limit cycles. In physical space the period doubling appears as the shedding of two distinct types of vorteX filaments. Further increases of the Reynolds number result in a cascade of period-doubling bifurcations, which create a chaotic state in the flow at a Reynolds number of about 500. The flow is characterized by broadband power spectra, and the appearance intermittent phenomena. It is concluded that the wake undergoes transit

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

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

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

GEOMETRIC CHARACTERIZATION OF STATES IN 2-DIMENSIONAL QUANTUM-GRAVITY

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MODERN PHYSICS LETTERS A 1990年第12期5卷 965-977页

作者： AGISHTEIN, ME JACOBS, L MIGDAL, AA RICHARDSON, JL Program in Applied and Computational Mathematics Princeton University Fine Hall Princeton NJ 08544 USA Center for Theoretical Physics 6-403 Massachusetts Institute of Technology Cambridge MA 02139 USA Work supported in part by the U. S. Department of Energy (D.O.E.) under contract #DE-AC02-76ER03069. Department of Physics and Program in Applied and Computational Mathematics Princeton University Fine Hall Princeton NJ 08544 USA Thinking Machines Corporation 245 First Street Cambridge MA 02142 USA

We report first results of a large-scale simulation of two-dimensional quantum gravity using the dynamical triangulation model for systems of up to sixteen thousand triangles. Our results for the internal geometry show an unexpectedly complicated behavior of the internal volume as function of the internal radius. A simple fractal characterization is inadequate to describe the geometry of the states in the system.

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LATTICE-VALUED BOTTLENECK DUALITY

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arXiv 2024年

作者： Ghrist, Robert Gould, Julian Lopez, Miguel Department of Mathematics Department of Electrical & Systems Engineering and Program in Applied Mathematics & Computational Science University of Pennsylvania United States

This note reformulates certain classical combinatorial duality theorems in the context of order lattices. For source-target networks, we generalize bottleneck path-cut and flow-cut duality results to edges with capacities in a distributive lattice. For posets, we generalize a bottleneck version of Dilworth’s theorem, again weighted in a distributive lattice. These results are applicable to a wide array of non-numerical network flow problems, as shown. All results, proofs, and applications were created in collaboration with AI language models. An appendix documents their role and *** Codes 06D05, 90C35, 06A07 © 2024, CC BY.

关键词： Combinatorial optimization

Featurized bidirectional GAN: Adversarial defense via adversarially learned semantic inference

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arXiv 2018年

作者： Bao, Ruying Liang, Sihang Wang, Qingcan Program in Applied and Computational Mathematics Princeton University Department of Physics Princeton University

Deep neural networks have been demonstrated to be vulnerable to adversarial attacks, where small perturbations intentionally added to the original inputs can fool the classifier. In this paper, we propose a defense method, Featurized Bidirectional Generative Adversarial Networks (FBGAN), to extract the semantic features of the input and filter the non-semantic perturbation. FBGAN is pre-trained on the clean dataset in an unsupervised manner, adversarially learning a bidirectional mapping between the high-dimensional data space and the low-dimensional semantic space;also mutual information is applied to disentangle the semantically meaningful features. After the bidirectional mapping, the adversarial data can be reconstructed to denoised data, which could be fed into any pre-trained classifier. We empirically show the quality of reconstruction images and the effectiveness of defense. Copyright © 2018, The Authors. All rights reserved.

关键词： Generative adversarial networks

Topological Classification of Insulators: II. Quasi-Two-Dimensional Locality

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arXiv 2024年

作者： Chung, Jui-Hui Shapiro, Jacob Program in Applied and Computational Mathematics Princeton University United States Department of Mathematics Princeton University United States

We provide an alternative characterization of two-dimensional locality (necessary e.g. to define the Hall conductivity of a Fermi projection) using the spectral projections of the Laughlin flux operator. Using this abstract characterization, we define generalizations of this locality, which we term quasi-2D. We go on to calculate the path-connected components of spaces of unitaries or orthogonal projections which are quasi-2D-local and find a starkly different behavior compared with the actual 2D column of the Kitaev table, exhibiting e.g., in the unitary chiral case, infinitely many Z-valued indices. Copyright © 2024, The Authors. All rights reserved.

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The quenching-activation behavior of the gradient descent dynamics for two-layer neural network models

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arXiv 2020年

作者： Ma, Chao Wu, Lei Weinan, E. Department of Mathematics Princeton University Program in Applied and Computational Mathematics Princeton University Institute of Big Data Research

A numerical and phenomenological study of the gradient descent (GD) algorithm for training two-layer neural network models is carried out for different parameter regimes when the target function can be accurately approximated by a relatively small number of neurons. It is found that for Xavier-like initialization, there are two distinctive phases in the dynamic behavior of GD in the under-parametrized regime: An early phase in which the GD dynamics follows closely that of the corresponding random feature model and the neurons are effectively quenched, followed by a late phase in which the neurons are divided into two groups: a group of a few "activated" neurons that dominate the dynamics and a group of background (or "quenched") neurons that support the continued activation and deactivation process. This neural network-like behavior is continued into the mildly over-parametrized regime, where it undergoes a transition to a random feature-like behavior. The quenching-activation process seems to provide a clear mechanism for "implicit regularization". This is qualitatively different from the dynamics associated with the "mean-field" scaling where all neurons participate equally and there does not appear to be qualitative changes when the network parameters are changed. Copyright © 2020, The Authors. All rights reserved.

关键词： Gradient methods

Neural network representation of electronic structure from ab initio molecular dynamics

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Science Bulletin 2022年第1期67卷 29-37,M0003页

作者： Qiangqiang Gu Linfeng Zhang Ji Feng International Center for Quantum Materials School of PhysicsPeking UniversityBeijing 100871China Department of Mathematics and Program in Applied and Computational Mathematics Princeton UniversityPrincetonNJ 08544USA Collaborative Innovation Center of Quantum Matter Beijing 100871China

Despite their rich information content,electronic structure data amassed at high volumes in ab initio molecular dynamics simulations are generally *** introduce a transferable high-fidelity neural network representation of such data in the form of tight-binding Hamiltonians for crystalline *** predictive representation of ab initio electronic structure,combined with machinelearning boosted molecular dynamics,enables efficient and accurate electronic evolution and *** it is applied to a one-dimension charge-density wave material,carbyne,we are able to compute the spectral function and optical conductivity in the canonical *** spectral functions evaluated during soliton-antisoliton pair annihilation process reveal significant renormalization of low-energy edge modes due to retarded electron-lattice coupling beyond the Born-Oppenheimer *** availability of an efficient and reusable surrogate model for the electronic structure dynamical system will enable calculating many interesting physical properties,paving the way to previously inaccessible or challenging avenues in materials modeling.

关键词： Neural network Tight-binding model Electronic structure ab initio molecular dynamics