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Essentially Commuting with a Unitary

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arXiv 2025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

Let R be a unitary operator whose spectrum is the circle. We show that the set of unitaries U which essentially commute with R (i.e., [U, R] ≡ UR − RU is compact) is path-connected. Moreover, we also calculate the set of path-connected components of the orthogonal projections which essentially commute with R and obey a non-triviality condition, and prove it is bijective with Z. Copyright © 2025, The Authors. All rights reserved.

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Thermal disorder and phonon softening in the ferroelectric phase transition of lead titanate

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Physical Review B 2025年第9期111卷 094113-094113页

作者： Pinchen Xie Yixiao Chen Weinan E Roberto Car Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA AI for Science Institute Beijing China and Center for Machine Learning Research and School of Mathematical Sciences Peking University Beijing China

We report a molecular dynamics study of ab initio quality of the ferroelectric phase transition in crystalline PbTiO3. We model anharmonicity accurately in terms of potential energy and polarization surfaces trained on density functional theory data with modern machine learning techniques. Our simulations demonstrate that the transition has a strong order-disorder character, in agreement with diffraction experiments, and provide fresh insight into the approach to equilibrium across the phase transition. We find that the emergence and disappearance of the macroscopic polarization is driven by dipolar switching at the nanometer scale. We also computed the infrared optical absorption spectra in both the ferroelectric and the paraelectric phases, finding good agreement with the experimental Raman frequencies. Often, the almost ideal displacive character of the soft mode detected by Raman scattering in the paraelectric phase has been contrasted with the order-disorder character of the transition suggested by diffraction experiments. We settle this issue by showing that the soft mode coexists with a strong Debye relaxation associated with thermal disordering of the dipoles. The Debye relaxation feature is centered at zero frequency and appears near the transition temperature in both the ferroelectric and the paraelectric phases.

关键词： Ferroelectricity

Revealing Actual Viscoelastic Relaxation Times in Capillary Breakup

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arXiv 2025年

作者： Hu, Nan Hwang, Jonghyun Ruangkriengsin, Tachin Stone, Howard A. Department of Mechanical and Aerospace Engineering Princeton University NJ08544 United States Program in Applied and Computational Mathematics Princeton University NJ08544 United States

We use experiments and theory to elucidate the size effect in capillary breakup rheometry, where pre-stretching in the visco-capillary stage causes the apparent relaxation time to be consistently smaller than the actual value. We propose a method accounting for both the experimental size and the finite extensibility of polymers to extract the actual relaxation time. A phase diagram characterizes the expected measurement variability and delineates scaling law conditions. The results refine capillary breakup rheometry for viscoelastic fluids and advance the understanding of breakup dynamics across scales. Copyright © 2025, The Authors. All rights reserved.

关键词： Relaxation time

Inferring System and Optimal Control Parameters of Closed-Loop Systems from Partial Observations

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arXiv 2025年

作者： Geadah, Victor Arbelaiz, Juncal Ritz, Harrison Daw, Nathaniel D. Cohen, Jonathan D. Pillow, Jonathan W. Program in Applied and Computational Mathematics Princeton University PrincetonNJ United States Department of Mechanical and Aerospace Engineering Princeton University PrincetonNJ United States Princeton Neuroscience Institute Princeton University PrincetonNJ United States

We consider the joint problem of system identification and inverse optimal control for discrete-time stochastic Linear Quadratic Regulators. We analyze finite and infinite time horizons in a partially observed setting, where the state is observed noisily. To recover closed-loop system parameters, we develop inference methods based on probabilistic state-space model (SSM) techniques. First, we show that the system parameters exhibit non-identifiability in the infinite-horizon from closed-loop measurements, and we provide exact and numerical methods to disentangle the parameters. Second, to improve parameter identifiability, we show that we can further enhance recovery by either (1) incorporating additional partial measurements of the control signals or (2) moving to the finite-horizon setting. We further illustrate the performance of our methodology through numerical examples. © 2025, CC BY.

关键词： Stochastic systems

Three-dimensional construction of hyperuniform, nonhyperuniform, and antihyperuniform disordered heterogeneous materials and their transport properties via spectral density functions

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Physical Review E 2025年第3期111卷 035310-035310页

作者： Wenlong Shi Yang Jiao Salvatore Torquato Materials Science and Engineering Arizona State University Tempe Arizona 85287 USA Department of Physics Arizona State University Tempe Arizona 85287 USA Department of Chemistry Princeton University Princeton New Jersey 08544 USA Department of Physics Princeton University Princeton New Jersey 08544 USA Princeton Institute of Materials Princeton University Princeton New Jersey 08544 USA Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA

Rigorous theories connecting physical properties of a heterogeneous material to its microstructure offer a promising avenue to guide the computational material design and optimization. The spectral density function χ̃V(k), which can be obtained experimentally from scattering data, enables accurate determination of various transport and wave propagation characteristics, including the time-dependent diffusion spreadability S(t) and effective dynamic dielectric constant εe for electromagnetic wave propagation. Moreover, χ̃V(k) determines rigorous upper bounds on the fluid permeability K. Given the importance of χ̃V(k), we present here an efficient Fourier-space based computational framework to construct three-dimensional (3D) statistically isotropic two-phase heterogeneous materials corresponding to targeted spectral density functions. In particular, we employ a variety of analytical functional forms for χ̃V(k) that satisfy all known necessary conditions to construct disordered stealthy hyperuniform, standard hyperuniform, nonhyperuniform, and antihyperuniform two-phase heterogeneous material systems at varying phase volume fractions. We show that by tuning the correlations in the system across length scales via the targeted functions, one can generate a rich spectrum of distinct structures within each of the above classes of materials. Importantly, we present the first realization of antihyperuniform two-phase heterogeneous materials in 3D, which are characterized by autocovariance function χV(r) with a power-law tail, resulting in microstructures that contain clusters of dramatically different sizes and morphologies. We also determine the diffusion spreadability S(t) and estimate the fluid permeability K associated with all of the constructed materials directly from the corresponding spectral densities. Although it is well established that the long-time asymptotic scaling behavior of S(t) only depends on the functional form of χ̃V(k), with the stealthy hyperuniform a

关键词： Spectral density

Effective Transport and Mechanical Properties of Two-Phase Materials Across the Order-Disorder Spectrum

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SSRN

SSRN 2025年

作者： Skolnick, Murray 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 Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

Two-phase heterogeneous materials arise in a plethora of natural and synthetic situations, such as alloys, composites, geological media, complex fluids, and biological media, exhibit a wide-variety of microstructures, and thus display a broad-spectrum of effective physical properties. Elucidating how microstructural details (e.g., specific surface, phase volume fractions, geometries, and topologies) influence two-phase materials’ effective transport and mechanical properties enhances our fundamental understanding of microstructure-property relationships, and is of great practical use in the design of structural and functional materials. Here, we compute the three-point microstructural parameters ζ2 and η2 for phase 2 to evaluate bounds and accurate approximation formulas on the effective thermal/electrical conductivities σe as well as the bulk Ke and shear Ge moduli of a wide class of disordered 2D and 3D material microstructures;including various dispersions of hard and penetrable particles, certain amorphous dispersions, and networks. These parameters are determined using a Monte Carlo integration scheme that incorporates certain algorithmic enhancements that improve upon the accuracy and performance of previous methods. Our results reveal that ζ2 and η2 are sensitive to the phase-connectedness properties of microstructures. Using numerical simulations and rigorous approximations that incorporate ζ2 and η2, we show that certain disordered microstructures nearly realize the Hashin-Shtrikman lower and upper bounds on σe, Ke, and Ge across volume-fractions. We also describe how our algorithm can be used in inverse methodologies to realize ordered and disordered materials with desirable effective physical properties by targeting specific values of ζ2 and η2, and hence aid in materials by design. © 2025, The Authors. All rights reserved.

关键词： Inverse problems

Can One Hear the Shape of a Crystal?

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arXiv 2025年

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

Isospectrality is a general fundamental concept often involving whether various operators can have identical spectra, i.e., the same set of eigenvalues. In the context of the Laplacian operator, the famous question "Can one hear the shape of a drum?" concerns whether different shaped drums can have the same vibrational modes. The isospectrality of a lattice in d-dimensional Euclidean space Rd is a tantamount to whether it is uniquely determined by its theta series, i.e., the radial distribution function g2(r). While much is known about the isospectrality of Bravais lattices across dimensions, little is known about this question of more general crystal (periodic) structures with an n-particle basis (n ≥ 2). Here, we ask, What is nmin(d), the minimum value of n for inequivalent (i.e., unrelated by isometric symmetries) crystals with the same theta function in space dimension d? To answer these questions, we use rigorous methods as well as a precise numerical algorithm that enables us to determine the minimum multi-particle basis of inequivalent isospectral crystals. Our algorithm identifies isospectral 4-, 3- and 2-particle bases in one, two and three spatial dimensions, respectively. For many of these isospectral crystals, we rigorously show that they indeed possess identical g2(r) up to infinite r. Based on our analyses, we conjecture that nmin(d) = 4, 3, 2 for d = 1, 2, 3, respectively. The identification of isospectral crystals enables one to study the degeneracy of the ground-state under the action of isotropic pair potentials. Indeed, using inverse statistical-mechanical techniques, we find an isotropic pair potential whose low-temperature configurations in two dimensions obtained via simulated annealing can lead to both of two isospectral crystal structures with n = 3, the proportion of which can be controlled by the cooling rate. Our findings provide general insights into the structural and ground-state degeneracies of crystal structures as determined by radia

关键词： Eigenvalues and eigenfunctions

FLUID-STRUCTURE INTERACTION WITH POROUS MEDIA: THE BEAVER-JOSEPH CONDITION IN THE STRONG SENSE

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arXiv 2025年

作者： Binz, Tim Hieber, Matthias Roy, Arnab Princeton University Program in Applied & Computational Mathematics Fine Hall Washington Road PrincetonNJ08544 United States Technische Universität Darmstadt Schloßgartenstraße 7 Darmstadt64289 Germany

This article considers fluid structure interaction describing the motion of a fluid contained in a porous medium. The fluid is modelled by Navier-Stokes equations and the coupling between fluid and the porous medium is described by the classical Beaver-Joseph or the Beaver-Joseph-Saffman interface condition. In contrast to previous work these conditions are investigated for the first time in the strong sense and it is shown that the coupled system admits a unique, global strong solution in critical spaces provided the data are small enough. Furthermore, a Serrin-type blow-up criterium is developed and higher regularity estimates at the interface are established, which say that the solution is even analytic provided the forces are so. Copyright © 2025, The Authors. All rights reserved.

关键词： Navier Stokes equations

Structural properties of hyperuniform Voronoi networks

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Physical Review E 2025年第3期111卷 034123-034123页

作者： Eli Newby Wenlong Shi Yang Jiao Reka Albert Salvatore Torquato Department of Physics Pennsylvania State University University Park Pennsylvania 16802 USA Materials Science and Engineering Arizona State University Tempe Arizona 85287 USA Department of Physics Arizona State University Tempe Arizona 85287 USA Department of Chemistry Princeton University Princeton New Jersey 08544 USA Department of Physics Princeton University Princeton New Jersey 08544 USA Princeton Institute of Materials Princeton University Princeton New Jersey 08544 USA Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA

Disordered hyperuniform many-particle systems are recently discovered exotic states of matter, characterized by the complete suppression of normalized infinite-wavelength density fluctuations, as in perfect crystals, while lacking conventional long-range order, as in liquids and glasses. In this work, we begin a program to quantify the structural properties of nonhyperuniform and hyperuniform networks. In particular, large two-dimensional (2D) Voronoi networks (graphs) containing approximately 10,000 nodes are created from a variety of different point configurations, including the antihyperuniform hyperplane intersection process (HIP), nonhyperuniform Poisson process, nonhyperuniform random sequential addition (RSA) saturated packing, and both non-stealthy and stealthy hyperuniform point processes. We carry out an extensive study of the Voronoi-cell area distribution of each of the networks by determining multiple metrics that characterize the distribution, including their average areas and corresponding variances as well as higher-order cumulants (i.e., skewness γ1 and excess kurtosis γ2). We show that the HIP distribution is far from Gaussian, as evidenced by a high skewness (γ1=3.16) and large positive excess kurtosis (γ2=16.2). The Poisson (with γ1=1.07 and γ2=1.79) and non-stealthy hyperuniform (with γ1=0.257 and γ2=0.0217) distributions are Gaussian-like distributions, since they exhibit a small but positive skewness and excess kurtosis. The RSA (with γ1=0.450 and γ2=−0.0384) and the highest stealthy hyperuniform distributions (with γ1=0.0272 and γ2=−0.0626) are also non-Gaussian because of their low skewness and negative excess kurtosis, which is diametrically opposite of the non-Gaussian behavior of the HIP. The fact that the cell-area distributions of large, finite-sized RSA and stealthy hyperuniform networks (e.g., with N≈10,000 nodes) are narrower, have larger peaks, and smaller tails than a Gaussian distribution implies that in the thermodynamic limit th

关键词： Gaussian distribution