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Exclusion Volumes of Convex Bodies in High Space Dimensions: Applications to Virial Coefficients and Continuum Percolation

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arXiv 2022年

作者： Torquato, Salvatore Jiao, Yang Department of Chemistry Princeton University PrincetonNJ08544 United States Department of Physics Princeton University PrincetonNJ08544 United States Princeton Institute of Materials Princeton University PrincetonNJ08544 United States Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Materials Science and Engineering Arizona State University TempeAZ85287 United States Department of Physics Arizona State University TempeAZ85287 United States

Using the concepts of mixed volumes and quermassintegrals of convex geometry, we derive an exact formula for the exclusion volume vex(K) for a general convex body K that applies in any space dimension. While our main interests concern the rotationally-averaged exclusion volume of a convex body with respect to another convex body, we also describe some results for the exclusion volumes for convex bodies with the same orientation. We show that the sphere minimizes the dimensionless exclusion volume vex(K)/v(K) among all convex bodies, whether randomly oriented or uniformly oriented, for any d, where v(K) is the volume of K. When the bodies have the same orientation, the simplex maximizes the dimensionless exclusion volume for any d with a large-d asymptotic scaling behavior of 22d/d3/2, which is to be contrasted with the corresponding scaling of 2d for the sphere. We present explicit formulas for quermassintegrals W0(K), . . ., Wd(K) for many different nonspherical convex bodies, including cubes, parallelepipeds, regular simplices, cross-polytopes, cylinders, spherocylinders, ellipsoids as well as lower-dimensional bodies, such as hyperplates and line segments. These results are utilized to determine the rotationally-averaged exclusion volume vex(K) for these convex-body shapes for dimensions 2 through 12. While the sphere is the shape possessing the minimal dimensionless exclusion volume, we show that, among the convex bodies considered that are sufficiently compact, the simplex possesses the maximal vex(K)/v(K) with a scaling behavior of 21.6618...d. Subsequently, we apply these results to determine the corresponding second virial coefficient B2(K) of the aforementioned hard hyperparticles. Our results are also applied to compute estimates of the continuum percolation threshold ηc derived previously by the authors for systems of identical overlapping convex bodies. We conjecture that overlapping spheres possess the maximal value of ηc among all identical nonzero-vol

关键词： Spheres

Neural Canonical Transformation with Symplectic Flows

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Physical Review X 2020年第2期10卷 021020-021020页

作者： Shuo-Hui Li Chen-Xiao Dong Linfeng Zhang Lei Wang Institute of Physics Chinese Academy of Sciences Beijing 100190 China University of Chinese Academy of Sciences Beijing 100049 China Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA Songshan Lake Materials Laboratory Dongguan Guangdong 523808 China

Canonical transformation plays a fundamental role in simplifying and solving classical Hamiltonian systems. Intriguingly, it has a natural correspondence to normalizing flows with a symplectic constraint. Building on this key insight, we design a neural canonical transformation approach to automatically identify independent slow collective variables in general physical systems and natural datasets. We present an efficient implementation of symplectic neural coordinate transformations and two ways to train the model based either on the Hamiltonian function or phase-space samples. The learned model maps physical variables onto an independent representation where collective modes with different frequencies are separated, which can be useful for various downstream tasks such as compression, prediction, control, and sampling. We demonstrate the ability of this method first by analyzing toy problems and then by applying it to real-world problems, such as identifying and interpolating slow collective modes of the alanine dipeptide molecule and MNIST database images.

关键词： Collective dynamics Hamiltonian systems Artificial neural networks Molecular dynamics

Stealthy and hyperuniform isotropic photonic bandgap structure in 3D

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arXiv 2024年

作者： Siedentop, Lukas Lui, Gianluc Maret, Georg Chaikin, Paul M. Steinhardt, Paul J. Torquato, Salvatore Keim, Peter Florescu, Marian Department of Physics University of Konstanz Konstanz78457 Germany Advanced Technology Institute Department of Physics University of Surrey Surrey GuildfordGU2 7XH United Kingdom Department of Physics New York University New YorkNY20012 United States Department of Physics Princeton University PrincetonNJ08544 United States Department of Chemistry Princeton University PrincetonNJ08544 United States Princeton Materials Institute Princeton University PrincetonNJ08544 United States Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Max-Planck-Institute for Dynamics and Self-Organization Göttingen37077 Germany Institute for the Dynamics of Complex Systems University of Göttingen Göttingen37077 Germany

In photonic crystals the propagation of light is governed by their photonic band structure, an ensemble of propagating states grouped into bands, separated by photonic band gaps. Due to discrete symmetries in spatially strictly periodic dielectric structures their photonic band structure is intrinsically anisotropic. However, for many applications, such as manufacturing artificial structural color materials or developing photonic computing devices, but also for the fundamental understanding of light-matter interactions, it is of major interest to seek materials with long range non-periodic dielectric structures which allow the formation of isotropic photonic band gaps. Here, we report the first ever 3D isotropic photonic band gap for an optimized disordered stealthy hyperuniform structure for microwaves. The transmission spectra are directly compared to a diamond pattern and an amorphous structure with similar node density. The band structure is measured experimentally for all three microwave structures, manufactured by 3D-Laser-printing for meta-materials with refractive index up to n = 2.1. Results agree well with finite-difference-time-domain numerical investigations and a priori calculations of the band-gap for the hyperuniform structure: the diamond structure shows gaps but being anisotropic as expected, the stealthy hyperuniform pattern shows an isotropic gap of very similar magnitude, while the amorphous structure does not show a gap at all. The centimeter scaled microwave structures may serve as prototypes for micrometer scaled structures with bandgaps in the technologically very interesting region of infrared (IR). Copyright © 2024, The Authors. All rights reserved.

关键词： Anisotropy

Reducing bias and variance for CTF estimation in single particle cryo-EM

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arXiv 2019年

作者： Heimowitz, Ayelet Andén, Joakim Singer, Amit Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Center for Computational Mathematics Flatiron Institute New YorkNY10010 United States Program in Applied and Computational Mathematics Department of Mathematics Princeton University PrincetonNJ08544 United States

When using an electron microscope for imaging of particles embedded in vitreous ice, the recorded image, or micrograph, is a significantly degraded version of the tomographic projection of the sample. Apart from noise, the image is affected by the optical configuration of the microscope. This transformation is typically modeled as a convolution with a point spread function. The Fourier transform of this function, known as the contrast transfer function (CTF), is oscillatory, attenuating and amplifying different frequency bands, and sometimes flipping their signs. High-resolution reconstruction requires this CTF to be accounted for, but as its form depends on experimental parameters, it must first be estimated from the micrograph. We present a new method for CTF estimation based on multitaper methods, which reduces bias and variance in the estimate. We also use known properties of the CTF and the background noise power spectrum to further reduce the variance through background subtraction and steerable basis projection. We show that the resulting power spectrum estimates better capture the zero-crossings of the CTF and yield accurate CTF estimates on several experimental micrographs. Copyright © 2019, The Authors. All rights reserved.

关键词： Linear programming

A Nodal Immersed Finite Element-Finite Difference Method

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arXiv 2021年

作者： Wells, David R. Vadala-Roth, Ben Lee, Jae H. Griffith, Boyce E. Department of Mathematics University of North Carolina Chapel HillNC United States Westborough MA United States Department of Mechanical Engineering Institute for Computational Medicine Johns Hopkins University BaltimoreMD United States Department of Mathematics Applied Physical Sciences and Biomedical Engineering University of North Carolina Chapel HillNC United States Carolina Center for Interdisciplinary Applied Mathematics University of North Carolina Chapel HillNC United States Computational Medicine Program University of North Carolina Chapel HillNC United States McAllister Heart Institute University of North Carolina Chapel HillNC United States Center for Drug Evaluation and Research U.S. Food and Drug Administration Silver SpringMD United States

The immersed finite element-finite difference (IFED) method is a computational approach to modeling interactions between a fluid and an immersed structure. The IFED method uses a finite element (FE) method to approximate the stresses, forces, and structural deformations on a structural mesh and a finite difference (FD) method to approximate the momentum and enforce the incompressibility of the entire fluid-structure system on a Cartesian grid. The fundamental approach used by this method follows the immersed boundary framework for modeling fluid-structure interaction (FSI), in which a force spreading operator prolongs structural forces to a Cartesian grid, and a velocity interpolation operator restricts a velocity field defined on that grid back onto the structural mesh. With an FE structural mechanics framework, force spreading first requires that the force itself be projected onto the finite element space. Similarly, velocity interpolation requires projecting velocity data onto the FE basis functions. Consequently, evaluating either coupling operator requires solving a matrix equation at every time step. Mass lumping, in which the projection matrices are replaced by diagonal approximations, has the potential to accelerate this method considerably. This paper provides both numerical and computational analyses of the effects of this replacement for evaluating the force projection and for the IFED coupling operators. Constructing the coupling operators also requires determining the locations on the structure mesh where the forces and velocities are sampled. Here we show that sampling the forces and velocities at the nodes of the structural mesh is equivalent to using lumped mass matrices in the IFED coupling operators. A key theoretical result of our analysis is that if both of these approaches are used together, the IFED method permits the use of lumped mass matrices derived from nodal quadrature rules for any standard interpolatory element. This is different from s

关键词： Fluid structure interaction

Multi-target detection with an arbitrary spacing distribution

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arXiv 2019年

作者： Lan, Ti-Yen Bendory, Tamir Boumal, Nicolas Singer, Amit Program in Applied and Computational Mathematics Mathematics Department Princeton University PrincetonNJ08544 United States

Motivated by the structure reconstruction problem in cryo-electron microscopy, we consider the multi-target detection model, where multiple copies of a target signal occur at unknown locations in a long measurement, further corrupted by additive Gaussian noise. At low noise levels, one can easily detect the signal occurrences and estimate the signal by averaging. However, in the presence of high noise, which is the focus of this paper, detection is impossible. Here, we propose two approaches—autocorrelation analysis and an approximate expectation maximization algorithm—to reconstruct the signal without the need to detect signal occurrences in the measurement. In particular, our methods apply to an arbitrary spacing distribution of signal occurrences. We demonstrate reconstructions with synthetic data and empirically show that the sample complexity of both methods scales as SNR−3 in the low SNR regime. Copyright © 2019, The Authors. All rights reserved.

关键词： Convolution

Critical pore radius and transport properties of disordered hard- and overlapping-sphere models

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Physical Review E 2021年第1期104卷 014127-014127页

作者： Michael A. Klatt Robert M. Ziff Salvatore Torquato Department of Physics Princeton University Princeton New Jersey 08544 USA Institut für Theoretische Physik FAU Erlangen-Nürnberg Staudtstr. 7 91058 Erlangen Germany Center for the Study of Complex Systems and Department of Chemical Engineering University of Michigan Ann Arbor Michigan 48109 USA Department of Chemistry Princeton Institute for the Science and Technology of Materials and Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA

Transport properties of porous media are intimately linked to their pore-space microstructures. We quantify geometrical and topological descriptors of the pore space of certain disordered and ordered distributions of spheres, including pore-size functions and the critical pore radius δc. We focus on models of porous media derived from maximally random jammed sphere packings, overlapping spheres, equilibrium hard spheres, quantizer sphere packings, and crystalline sphere packings. For precise estimates of the percolation thresholds, we use a strict relation of the void percolation around sphere configurations to weighted bond percolation on the corresponding Voronoi networks. We use the Newman-Ziff algorithm to determine the percolation threshold using universal properties of the cluster size distribution. The critical pore radius δc is often used as the key characteristic length scale that determines the fluid permeability k. A recent study [Torquato, Adv. Wat. Resour. 140, 103565 (2020)] suggested for porous media with a well-connected pore space an alternative estimate of k based on the second moment of the pore size 〈δ2〉, which is easier to determine than δc. Here, we compare δc to the second moment of the pore size 〈δ2〉, and indeed confirm that, for all porosities and all models considered, δc2 is to a good approximation proportional to 〈δ2〉. However, unlike 〈δ2〉, the permeability estimate based on δc2 does not predict the correct ranking of k for our models. Thus, we confirm 〈δ2〉 to be a promising candidate for convenient and reliable estimates of the fluid permeability for porous media with a well-connected pore space. Moreover, we compare the fluid permeability of our models with varying degrees of order, as measured by the τ order metric. We find that (effectively) hyperuniform models tend to have lower values of k than their nonhyperuniform counterparts. Our findings could facilitate the design of porous media with desirable transport properties via targete

关键词： Geophysical networks Materials Patterns in complex systems

Spectral Top-Down Recovery of Latent Tree Models

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arXiv 2021年

作者： Aizenbud, Yariv Jaffe, Ariel Wang, Meng Hu, Amber Amsel, Noah Nadler, Boaz Chang, Joseph T. Kluger, Yuval Program in Applied Mathematics Yale University New HavenCT06511 United States Department of Computer Science Weizmann Institute of Science Rehovot76100 Israel Department of Statistics Yale University New HavenCT06520 United States Interdepartmental Program in Computational Biology and Bioinformatics Yale University New HavenCT06511 United States Department of Pathology Yale University New HavenCT06511 United States

Modeling the distribution of high dimensional data by a latent tree graphical model is a prevalent approach in multiple scientific domains. A common task is to infer the underlying tree structure, given only observations of its terminal nodes. Many algorithms for tree recovery are computationally intensive, which limits their applicability to trees of moderate size. For large trees, a common approach, termed divide-and-conquer, is to recover the tree structure in two steps. First, recover the structure separately of multiple, possibly random subsets of the terminal nodes. Second, merge the resulting subtrees to form a full tree. Here, we develop Spectral Top-Down Recovery (STDR), a deterministic divide-and-conquer approach to infer large latent tree models. Unlike previous methods, STDR partitions the terminal nodes in a non random way, based on the Fiedler vector of a suitable Laplacian matrix related to the observed nodes. We prove that under certain conditions, this partitioning is consistent with the tree structure. This, in turn, leads to a significantly simpler merging procedure of the small subtrees. We prove that STDR is statistically consistent and bound the number of samples required to accurately recover the tree with high probability. Using simulated data from several common tree models in phylogenetics, we demonstrate that STDR has a significant advantage in terms of runtime, with improved or similar accuracy. Copyright © 2021, The Authors. All rights reserved.

关键词： Recovery

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