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Disordered Heterogeneous Universe: Galaxy Distribution and Clustering across Length Scales

Physical Review X 2023年第1期13卷 011038-011038页

作者： Oliver H. E. Philcox Salvatore Torquato Department of Astrophysical Sciences Princeton University Princeton New Jersey 08540 USA School of Natural Sciences Institute for Advanced Study 1 Einstein Drive Princeton New Jersey 08540 USA Center for Theoretical Physics Department of Physics Columbia University New York New York 10027 USA Simons Foundation New York New York 10010 USA Department of Chemistry Department of Physics Princeton Materials Institute and Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08540 USA

The studies of disordered heterogeneous media and galaxy cosmology share a common goal: analyzing the disordered distribution of particles and/or building blocks at microscales to predict physical properties of the medium at macroscales, whether it be a liquid, colloidal suspension, composite material, galaxy cluster, or entire Universe. The theory of disordered heterogeneous media provides an array of theoretical and computational techniques to characterize a wide class of complex material microstructures. In this work, we apply them to describe the disordered distributions of galaxies obtained from recent suites of dark matter simulations. We focus on the determination of lower-order correlation functions, void and particle nearest-neighbor functions, certain cluster statistics, pair-connectedness functions, percolation properties, and a scalar order metric to quantify the degree of order. Compared to analogous homogeneous Poisson and typical disordered systems, the cosmological simulations exhibit enhanced large-scale clustering and longer tails in the void and particle nearest-neighbor functions, due to the presence of quasi-long-range correlations imprinted by early Universe physics, with a minimum particle separation far below the mean nearest-neighbor distance. On large scales, the system appears hyperuniform, as a result of primordial density fluctuations, while on the smallest scales, the system becomes almost antihyperuniform, as evidenced by its number variance. Additionally, via a finite-scaling analysis, we compute the percolation threshold of the galaxy catalogs, finding this to be significantly lower than for Poisson realizations (at reduced density ηc=0.25 in our fiducial analysis compared to ηc=0.34), with strong dependence on the mean density; this is consistent with the observation that the galaxy distribution contains voids of up to 50% larger radius. However, the two sets of simulations appear to share the same fractal dimension on scales much l

关键词： Large scale structure of the Universe

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

arXiv

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arXiv 2024年

作者： Wang, Haina Samajdar, Rhine Torquato, Salvatore Department of Chemistry Princeton University PrincetonNJ08544 United States Department of Physics Princeton University PrincetonNJ08544 United States Princeton Center for Theoretical Science Princeton University PrincetonNJ08544 United States Princeton Materials Institute Princeton University PrincetonNJ08540 United States Program in Applied and Computational Mathematics PrincetonNJ08544 United States

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 typical liquids, i.e., the structure factor obeys the scaling relation S(k) ∼ Bkα with B, α > 0 in the limit k → 0. Ground-state 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 work, we systematically address these questions by deriving the analytical small-k behaviors (and associatedly, α and B) of the total and spin-resolved structure factors of quasi-1D, 2D, and 3D 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 work thus reveals that highly unusual hyperuniform and multihyperuniform states can be achieved in simple fermionic systems and paves the way for harnessing the unique hyperuniform

关键词： Density of liquids

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Hyperuniformity Classes of Quasiperiodic Tilings via Diffusion Spreadability

arXiv

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arXiv 2024年

作者： Hitin-Bialus, Adam Maher, Charles Emmett Steinhardt, Paul J. Torquato, Salvatore Department of Physics Princeton University PrincetonNJ08544 United States Department of Chemistry Princeton University PrincetonNJ08544 United States Jefferson Physical Laboratory Harvard University CambridgeMA02138 United States Princeton Institute of Materials Princeton University PrincetonNJ08544 United States Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

Hyperuniform point patterns can be classified by the hyperuniformity scaling exponent α > 0, that characterizes the power-law scaling behavior of the structure factor S(k) as a function of wavenumber k ≡ |k| in the vicinity of the origin, e.g., S(k) ∼ |k|α in cases where S(k) varies continuously with k as k → 0. In this paper, we show that the spreadability is an effective method for determining α for quasiperiodic systems where S(k) is discontinuous and consists of a dense set of Bragg peaks. It has been shown in [Torquato, Phys. Rev. E 104, 054102 (2021)] that, for media with finite α, the long-time behavior of the excess spreadability S(∞) − S(t) can be fit to a power law of the form ∼ t−(d−α)/2, where d is the space dimension, to accurately extract α for the continuous case. We first transform quasiperiodic and limit-periodic point patterns into two-phase media by mapping them onto packings of identical nonoverlapping disks, where space interior to the disks represents one phase and the space in exterior to them represents the second phase. We then compute the spectral density χ˜V (k) of the packings, and finally compute and fit the long-time behavior of their excess spreadabilities. Specifically, we show that the excess spreadability can be used to accurately extract α for the 1D limit-periodic period doubling chain (α = 1) and the 1D quasicrystalline Fibonacci chain (α = 3) to within 0.02% of the analytically known exact results. Moreover, we obtain a value of α = 5.97 ± 0.06 for the 2D Penrose tiling, which had not been computed previously, and present plausible theoretical arguments strongly suggesting that α is exactly equal to 6. We also show that, due to the self-similarity of the structures examined here, one can truncate the small-k region of the scattering information used to compute the spreadability and obtain an accurate value of α, with a small deviation from the untruncated case that decreases as the system size increases. This strongly suggests t

关键词： Spectral density

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Composite B-Spline Regularized Delta Functions for the Immersed Boundary Method: Divergence-Free Interpolation and Gradient-Preserving Force Spreading

arXiv

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arXiv 2024年

作者： Gruninger, Cole Griffith, Boyce E. Department of Mathematics University North Carolina Chapel HillNC United States Department of Applied Physical Sciences University of North Carolina Chapel HillNC United States Department of 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 School of Medicine Chapel HillNC United States McAllister Heart Institute University of North Carolina School of Medicine Chapel HillNC United States

This paper presents an approach to enhance volume conservation in the immersed boundary (IB) method by employing regularized delta functions derived from composite B-splines. These delta functions are constructed using tensor product kernels, similar to the conventional IB method. However, the kernels are B-splines whose polynomial degree varies according to the normal and tangential directions of each velocity component. The conventional IB method, while effective for fluid-structure interaction applications, has long been challenged by poor volume conservation, particularly evident in simulations of pressurized, closed membranes. We demonstrate that composite B-spline regularized delta functions significantly enhance volume conservation through two complementary properties: they provide continuously divergence-free velocity interpolants and maintain the gradient character of forces corresponding to mean pressure jumps across interfaces. By correctly representing these forces as discrete gradients, they eliminate a key source of spurious flows that typically plague immersed boundary computations. Our approach maintains the local nature of the classical IB method, avoiding the computational overhead associated with the non-local Divergence-Free Immersed Boundary (DFIB) method’s construction of an explicit velocity potential which requires additional Poisson solves for interpolation and force spreading operations. Through a series of numerical experiments, we show that sufficiently regular composite B-spline kernels can maintain initial volumes to within machine precision. We provide a detailed analysis of the relationship between kernel regularity and the accuracy of force spreading and velocity interpolation operations. Our findings indicate that composite B-splines of at least C1 regularity produce results comparable to the DFIB method in dynamic simulations, with errors in volume conservation primarily dominated by truncation error of the employed time-stepping s

关键词： Delta functions

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Scaling limits of external multi-particle DLA on the plane and the supercooled stefan problem

arXiv

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arXiv 2021年

作者： Nadtochiy, Sergey Shkolnikov, Mykhaylo Zhang, Xiling Department of Applied Mathematics Illinois Institute of Technology ChicagoIL60616 United States ORFE Department Bendheim Center for Finance Program in Applied & Computational Mathematics Princeton University PrincetonNJ08544 United States

We consider (a variant of) the external multi-particle diffusion-limited aggregation (MDLA) process of Rosenstock and Marquardt on the plane. Based on the recent findings of [11], [10] in one space dimension it is natural to conjecture that the scaling limit of the growing aggregate in such a model is given by the growing solid phase in a suitable "probabilistic" formulation of the single-phase supercooled Stefan problem for the heat equation. To address this conjecture, we extend the probabilistic formulation from [10] to multiple space dimensions. We then show that the equation that characterizes the growth rate of the solid phase in the supercooled Stefan problem is satisfied by the scaling limit of the external MDLA process with an inequality, which can be strict in general. In the course of the proof, we establish two additional results interesting in their own right: (i) the stability of a "crossing property" of planar Brownian motion and (ii) a rigorous connection between the probabilistic solutions to the supercooled Stefan problem and its classical and weak *** Codes 82C22, 80A22, 60H30 Copyright © 2021, The Authors. All rights reserved.

关键词： Supercooling

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Artificial neural network approach for turbulence models: A local framework

引用

Physical Review Fluids 2021年第8期6卷 084612-084612页

作者： Chenyue Xie Xiangming Xiong Jianchun Wang Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA Department of Mechanics and Aerospace Engineering Southern University of Science and Technology Shenzhen 518055 People's Republic of China

A local artificial neural network (LANN) framework is developed for turbulence modeling. The Reynolds-averaged Navier-Stokes (RANS) unclosed terms are reconstructed by the artificial neural network based on the local coordinate system which is orthogonal to the curved walls. We verify the proposed model in the flows over periodic hills. The correlation coefficients of the RANS unclosed terms predicted by the LANN model can be made larger than 0.96 in an a priori analysis, and the relative error of the unclosed terms can be made smaller than 18%. In an a posteriori analysis, detailed comparisons are made on the results of RANS simulations using the LANN, global artificial neural network (GANN), Spalart-Allmaras (SA), and shear stress transport (SST) k−ω models. It is shown that the LANN model performs better than the GANN, SA, and SST k−ω models in the prediction of the average velocity, wall-shear stress, and average pressure, which gives the results that are essentially indistinguishable from the direct numerical simulation data. The LANN model trained at low Reynolds number, Re=2800, can be directly applied to the cases of high Reynolds numbers, Re=5600, 10 595, 19 000, and 37 000, with accurate predictions. Furthermore, the LANN model is verified for flows over periodic hills with varying slopes. These results suggest that the LANN framework has a great potential to be applied to complex turbulent flows with curved walls.

关键词： Compressible boundary layers Turbulence Artificial neural networks

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Sequential Markov Chain Monte Carlo for Lagrangian Data Assimilation with Applications to Unknown Data Locations

arXiv

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arXiv 2023年

作者： Ruzayqat, Hamza Beskos, Alexandros Crisan, Dan Jasra, Ajay Kantas, Nikolas Applied Mathematics and Computational Science Program Computer Electrical and Mathematical Sciences and Engineering Division King Abdullah University of Science and Technology Thuwal23955-6900 Saudi Arabia Department of Statistical Science University College London LondonWC1E 6BT United Kingdom Department of Mathematics Imperial College London LondonSW7 2AZ United Kingdom

We consider a class of high-dimensional spatial filtering problems, where the spatial locations of observations are unknown and driven by the partially observed hidden signal. This problem is exceptionally challenging as not only is high-dimensional, but the model for the signal yields longer-range time dependencies through the observation locations. Motivated by this model we revisit a lesser-known and provably convergent computational methodology from [4, 11, 28] that uses sequential Markov Chain Monte Carlo (MCMC) chains. We extend this methodology for data filtering problems with unknown observation locations. We benchmark our algorithms on Linear Gaussian state space models against competing ensemble methods and demonstrate a significant improvement in both execution speed and accuracy. Finally, we implement a realistic case study on a high-dimensional rotating shallow water model (of about 104 - 105 dimensions) with real and synthetic data. The data is provided by the National Oceanic and Atmospheric Administration (NOAA) and contains observations from ocean drifters in a domain of the Atlantic Ocean restricted to the longitude and latitude intervals [-51◦,-41◦], [17◦,27◦] *** Codes 62M20, 60G35, 60J20, 94A12, 93E11, 65C40 © 2023, CC BY.

关键词： Location

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Diffusion spreadability as a probe of the microstructure of complex media across length scales

arXiv

引用

arXiv 2021年

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

Understanding time-dependent diffusion processes in multiphase media is of great importance in physics, chemistry, materials science, petroleum engineering and biology. Consider the time-dependent problem of mass transfer of a solute between two phases and assume that the solute is initially distributed in one phase (phase 2) and absent from the other (phase 1). We desire the fraction of total solute present in phase 1 as a function of time, S(t), which we call the spreadability, since it is a measure of the spreadability of diffusion information as a function of time. We derive exact direct-space formulas for S(t) in any Euclidean space dimension d in terms of the autocovariance function as well as corresponding Fourier representations of S(t) in terms of the spectral density, which are especially useful when scattering information is available experimentally or theoretically. These are singular results because they are rare examples of mass transport problems where exact solutions are possible. We derive closed-form general formulas for the short- and long-time behaviors of the spreadability in terms of crucial small- and large-scale microstructural information, respectively. The long-time behavior of S(t) enables one to distinguish the entire spectrum of microstructures that span from hyperuniform to nonhyperuniform media. For hyperuniform media, disordered or not, we show that the "excess" spreadability, S(∞) − S(t), decays to its long-time behavior exponentially faster than that of any nonhyperuniform two-phase medium, the "slowest" being antihyperuniform media. The stealthy hyperuniform class is characterized by an excess spreadability with the fastest decay rate among all translationally invariant microstructures. We obtain exact results for S(t) for a variety of specific ordered and disordered model microstructures across dimensions that span from hyperuniform to antihyperuniform media. Moreover, we establish a remarkable connection between the spreadability

关键词： Microstructure

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Inferring solar differential rotation through normal-mode coupling using bayesian statistics

arXiv

引用

arXiv 2021年

作者： Kashyap, Samarth Ganesh Bharati Das, Srijan Hanasoge, Shravan M. Woodard, Martin F. Tromp, Jeroen Department of Astronomy and Astrophysics Tata Institute of Fundamental Research Mumbai India Department of Geosciences Princeton University PrincetonNJ United States NorthWest Research Associates Boulder Office 3380 Mitchell Lane BoulderCO United States Department of Geosciences and Program in Applied & Computational Mathematics Princeton University PrincetonNJ United States

Normal-mode helioseismic data analysis uses observed solar oscillation spectra to infer perturbations in the solar interior due to global and local-scale flows and structural asphericity. Differential rotation, the dominant global-scale axisymmetric perturbation, has been tightly constrained primarily using measurements of frequency splittings via "a-coefficients". However, the frequency-splitting formalism invokes the approximation that multiplets are isolated. This assumption is inaccurate for modes at high angular degrees. Analysing eigenfunction corrections, which respect cross coupling of modes across multiplets, is a more accurate approach. However, applying standard inversion techniques using these cross-spectral measurements yields a-coefficients with a significantly wider spread than the well-constrained results from frequency splittings. In this study, we apply Bayesian statistics to infer a-coefficients due to differential rotation from cross spectra for both f-modes and p-modes. We demonstrate that this technique works reasonably well for modes with angular degrees ` = 50 − 291. The inferred a3−coefficients are found to be within 1 nHz of the frequency splitting values for ` > 200. We also show that the technique fails at ` © 2021, CC BY.

关键词： Eigenvalues and eigenfunctions

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Ordered and disordered stealthy hyperuniform point patterns across spatial dimensions

arXiv

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arXiv 2024年

作者： Morse, Peter K. Steinhardt, Paul J. Torquato, Salvatore 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 Princeton Center for Theoretical Science Princeton University PrincetonNJ08544 United States Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

In previous work [Phys. Rev. X 5, 021020 (2015)], it was shown that stealthy hyperuniform systems can be regarded as hard spheres in Fourier-space in the sense that the the structure factor is exactly zero in a spherical region around the origin in analogy with the pair-correlation function of real-space hard spheres. In this work, we exploit this correspondence to confirm that the densest Fourier-space hard-sphere system is that of a Bravais lattice. This is in contrast to real-space hard-spheres, whose densest configuration is conjectured to be disordered. We also extend the virial series previously suggested for disordered stealthy hyperuniform systems to higher dimensions in order to predict spatial decorrelation as function of dimension. This prediction is then borne out by numerical simulations of disordered stealthy hyperuniform ground states in dimensions d = 2-8. © 2024, CC BY.

关键词： Ground state

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