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Designer spin systems via inverse statistical mechanics

Physical Review B 2013年第13期88卷 134104-134104页

作者： Robert A. DiStasio, Jr. Étienne Marcotte Roberto Car Frank H. Stillinger Salvatore Torquato Department of Chemistry Princeton University Princeton New Jersey 08544 USA Department of Physics Princeton University Princeton New Jersey 08544 USA Princeton Institute for the Science and Technology of Materials Princeton University Princeton New Jersey 08544 USA Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA

In this work, we extend recent inverse statistical-mechanical methods developed for many-particle systems to the case of spin systems. For simplicity, we focus in this initial study on the two-state Ising model with radial spin-spin interactions of finite range (i.e., extending beyond nearest-neighbor sites) on the square lattice under periodic boundary conditions. Our interest herein is to find the optimal set of shortest-range pair interactions within this family of Hamiltonians, whose corresponding ground state is a targeted spin configuration such that the difference in energies between the energetically closest competitor and the target is maximized. For an exhaustive list of competitors, this optimization problem is solved exactly using linear programming. The possible outcomes for a given target configuration can be organized into the following three solution classes: unique (nondegenerate) ground state (class I), degenerate ground states (class II), and solutions not contained in the previous two classes (class III). We have chosen to study a general family of striped-phase spin configurations comprised of alternating parallel bands of up and down spins of varying thicknesses and a general family of rectangular block checkerboard spin configurations with variable block size, which is a generalization of the classic antiferromagnetic Ising model. Our findings demonstrate that the structurally anisotropic striped phases, in which the thicknesses of up- and down-spin bands are equal, are unique ground states for isotropic short-ranged interactions. By contrast, virtually all of the block checkerboard targets are either degenerate or fall within class III solutions. The degenerate class II spin configurations are identified up to a certain block size. We also consider other target spin configurations with different degrees of global symmetries and order. Our investigation reveals that the solution class to which a target belongs depends sensitively on the nature

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A hybrid semi-Lagrangian cut cell method for advection-diffusion problems with robin boundary conditions in moving domains

arXiv

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arXiv 2021年

作者： Barrett, Aaron Fogelson, Aaron L. Griffith, Boyce E. Department of Mathematics University of Utah Salt Lake CityUT United States Departments of Mathematics and Bioengineering University of Utah Salt Lake CityUT United States Departments 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

We present a new discretization approach to advection-diffusion problems with Robin boundary conditions on complex, time-dependent domains. The method is based on second order cut cell finite volume methods introduced by Bochkov et al. [8] to discretize the Laplace operator and Robin boundary condition. To overcome the small cell problem, we use a splitting scheme along with a semi-Lagrangian method to treat advection. We demonstrate second order accuracy in the L1, L2, and L∞norms for both analytic test problems and numerical convergence studies. We also demonstrate the ability of the scheme to convert one chemical species to another across a moving boundary. © 2021, CC BY-SA.

关键词： Advection

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Designer spin systems via inverse statistical mechanics. II. Ground-state enumeration and classification

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Physical Review B 2013年第18期88卷 184432-184432页

作者： Étienne Marcotte Robert A. DiStasio, Jr. Frank H. Stillinger Salvatore Torquato Department of Physics Princeton University Princeton New Jersey 08544 USA Department of Chemistry Princeton University Princeton New Jersey 08544 USA Princeton Institute for the Science and Technology of Materials Princeton University Princeton New Jersey 08544 USA Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA

In the first paper of this series [DiStasio, Jr., Marcotte, Car, Stillinger, and Torquato, Phys. Rev. B 88, 134104 (2013)], we applied inverse statistical-mechanical techniques to study the extent to which targeted spin configurations on the square lattice can be ground states of finite-ranged radial spin-spin interactions. In this sequel, we enumerate all of the spin configurations within a unit cell on the one-dimensional integer lattice and the two-dimensional square lattice up to some modest size under periodic boundary conditions. We then classify these spin configurations into those that can or cannot be unique classical ground states of the aforementioned radial pair spin interactions and found the relative occurrences of these ground-state solution classes for different system sizes. As a result, we also determined the minimal radial extent of the spin-spin interaction potentials required to stabilize those configurations whose ground states are either unique or degenerate (i.e., those sharing the same radial spin-spin correlation function). This enumeration study has established that unique ground states are not limited to simple target configurations. However, we also found that many simple target spin configurations cannot be unique ground states.

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Robust Long-Term Growth Rate of Expected Utility for Leveraged ETFs

arXiv

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arXiv 2023年

作者： Leung, Tim Park, Hyungbin Yeo, Heejun Department of Applied Mathematics Computational Finance & Risk Management Program University of Washington SeattleWA98195 United States Department of Mathematical Sciences Research Institute of Mathematics Seoul National University 1 Gwanak-ro Gwanak-gu Seoul08826 Korea Republic of Department of Mathematical Sciences Seoul National University 1 Gwanak-ro Gwanak-gu Seoul08826 Korea Republic of

This paper analyzes the robust long-term growth rate of expected utility and expected return from holding a leveraged exchange-traded fund (LETF). When the Markovian model parameters in the reference asset are uncertain, the robust long-term growth rate is derived by analyzing the worst-case parameters among an uncertainty set. We compute the growth rate and describe the optimal leverage ratio maximizing the robust long-term growth rate. To achieve this, the worst-case parameters are analyzed by the comparison principle, and the growth rate of the worst-case is computed using the martingale extraction method. The robust long-term growth rates are obtained explicitly under a number of models for the reference asset, including the geometric Brownian motion (GBM), Cox–Ingersoll–Ross (CIR), 3/2, and Heston and 3/2 stochastic volatility models. Additionally, we demonstrate the impact of stochastic interest rates, such as the Vasicek and inverse GARCH short rate models. This paper is an extended work of Leung and Park [2017]. Copyright © 2023, The Authors. All rights reserved.

关键词： Growth rate

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Viscoelastic fluid flow in a slowly varying planar contraction: the role of finite extensibility on the pressure drop

arXiv

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arXiv 2024年

作者： Mahapatra, Bimalendu Ruangkriengsin, Tachin Stone, Howard A. Boyko, Evgeniy Faculty of Mechanical Engineering Technion – Israel Institute of Technology Haifa3200003 Israel Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Department of Mechanical and Aerospace Engineering Princeton University PrincetonNJ08544 United States

We analyze the steady viscoelastic fluid flow in slowly varying contracting channels of arbitrary shape and present a theory based on the lubrication approximation for calculating the flow rate–pressure drop relation at low and high Deborah (De) numbers. Unlike most prior theoretical studies leveraging the Oldroyd-B model, we describe the fluid viscoelasticity using a FENE-CR model and examine how the polymer chains’ finite extensibility impacts the pressure drop. We employ the low-Deborah-number lubrication analysis to provide analytical expressions for the pressure drop up to O(De4). We further consider the ultra-dilute limit and exploit a one-way coupling between the parabolic velocity and elastic stresses to calculate the pressure drop of the FENE-CR fluid for arbitrary values of the Deborah number. Such an approach allows us to elucidate elastic stress contributions governing the pressure drop variations and the effect of finite extensibility for all De. We validate our theoretical predictions with two-dimensional numerical simulations and find excellent agreement. We show that, at low Deborah numbers, the pressure drop of the FENE-CR fluid monotonically decreases with De, similar to the previous results for the Oldroyd-B and FENE-P fluids. However, at high Deborah numbers, in contrast to a linear decrease for the Oldroyd-B fluid, the pressure drop of the FENE-CR fluid exhibits a non-monotonic variation due to finite extensibility, first decreasing and then increasing with de. Nevertheless, even at sufficiently high Deborah numbers, the pressure drop of the FENE-CR fluid in the ultra-dilute and lubrication limits is lower than the corresponding Newtonian pressure drop. © 2024, CC BY.

关键词： Non Newtonian flow

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DeePN2: A deep learning-based non-Newtonian hydrodynamic model

arXiv

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arXiv 2021年

作者： Fang, Lidong Ge, Pei Zhang, Lei Weinan, E. Lei, Huan Department of Computational Mathematics Science and Engineering Michigan State University MI48824 United States School of Mathematical Sciences Institute of Natural Sciences and MOE-LSC Shanghai Jiao Tong University 800 Dongchuan Road Shanghai200240 China Center for Machine Learning Research School of Mathematical Sciences Peking University Beijing100871 China AI for Science Institute Beijing100080 China Department of Mathematics and Program in Applied and Computational Mathematics Princeton University NJ08544 United States Department of Statistics and Probability Michigan State University MI48824 United States

A long standing problem in the modeling of non-Newtonian hydrodynamics of polymeric flows is the availability of reliable and interpretable hydrodynamic models that faithfully encode the underlying micro-scale polymer dynamics. The main complication arises from the long polymer relaxation time, the complex molecular structure and heterogeneous interaction. DeePN2, a deep learning-based non-Newtonian hydrodynamic model, has been proposed and has shown some success in systematically passing the micro-scale structural mechanics information to the macro-scale hydrodynamics for suspensions with simple polymer conformation and bond potential. The model retains a multi-scaled nature by mapping the polymer configurations into a set of symmetry-preserving macro-scale features. The extended constitutive laws for these macro-scale features can be directly learned from the kinetics of their micro-scale counterparts. In this paper, we develop DeePN2 using more complex micro-structural models. We show that DeePN2 can faithfully capture the broadly overlooked viscoelastic differences arising from the specific molecular structural mechanics without human intervention. © 2021, CC BY.

关键词： Non Newtonian flow

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Deep potentials for materials science

材料展望（英文）

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材料展望（英文） 2022年第2期 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 AI for Science Institute BeijingPeople'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 Department of Mechanical Engineering The University of Hong KongHong KongHong Kong Special Administrative Region of China 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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Geometry-based data generation

arXiv

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arXiv 2018年

作者： Lindenbaum, Ofir Stanley, Jay S. Wolf, Guy Krishnaswamy, Smita Department of Genetics Yale University New HavenCT United States Applied Mathematics Program Yale University New HavenCT United States Computational Biology & Bioinformatics Program Yale University New HavenCT United States Department of Computer Science Yale University New HavenCT United States

We propose a new type of generative model of high-dimensional data that learns a manifold geometry of the data, rather than density, and can generate points evenly along this manifold. This is in contrast to existing generative models that represent data density, and are strongly affected by noise and other artifacts of data collection. We demonstrate how this approach corrects sampling biases and artifacts, and improves several downstream data analysis tasks such as clustering and classification. Finally, we demonstrate that this approach is especially useful in biology where, despite the advent of single-cell technologies, rare subpopulations and gene-interaction relationships are affected by biased sampling. We show that SUGAR can generate hypothetical populations and reveal intrinsic patterns and mutual-information relationships between genes on a single cell dataset of hematopoiesis. Copyright © 2018, The Authors. All rights reserved.

关键词： Clustering algorithms

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Minuscule corrections to near-surface solar internal rotation using mode-coupling

arXiv

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arXiv 2023年

作者： Das, Srijan Bharati Kashyap, Samarth G. Oktay, Deniz Hanasoge, Shravan M. Tromp, Jeroen Department of Geosciences Princeton University PrincetonNJ United States Department of Astronomy and Astrophysics Tata Institute of Fundamental Research Mumbai India Department of Computer Science Princeton University PrincetonNJ United States Department of Geosciences and Program in Applied & Computational Mathematics Princeton University PrincetonNJ United States

The observed solar oscillation spectrum is influenced by internal perturbations such as flows and structural asphericities. These features induce splitting of characteristic frequencies and distort the resonant-mode eigenfunctions. Global axisymmertric flow — differential rotation — is a very prominent perturbation. Tightly constrained rotation profiles as a function of latitude and radius are products of established helioseismic pipelines that use observed Dopplergrams to generate frequency-splitting measurements at high precision. However, the inference of rotation using frequency-splittings do not consider the effect of mode-coupling. This approximation worsens for high-angular-degree modes, as they become increasingly proximal in frequency. Since modes with high angular degrees probe the near-surface layers of the Sun, inversions considering coupled modes could potentially lead to more accurate estimates of rotation very close to the surface. In order to investigate if this is indeed the case, we perform inversions for solar differential rotation, considering coupling of modes for angular degrees 160 ≤ ` ≤ 300 in the surface gravity f-branch and first-overtone p modes. In keeping with the character of mode coupling, we carry out a non-linear inversion using an eigenvalue solver. Differences in inverted profiles for frequency splitting measurements from MDI and HMI are compared and discussed. We find that corrections to the near-surface differential rotation profile, when accounting for mode-coupling effects, are smaller than 0.003 nHz and hence are insignificant. These minuscule corrections are found to be correlated with the solar cycle. We also present corrections to even-order splitting coefficients, which could consequently impact inversions for structure and magnetic fields. © 2023, CC BY.

关键词： Rotation

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Consensus dynamics and coherence in hierarchical small-world networks

arXiv

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arXiv 2023年

作者： Liao, Yunhua Maama, Mohamed Aziz-Alaoui, M.A. Department of Mathematics Hunan University of Technology and Business Hunan Changsha410205 China Key Laboratory of Hunan Province for Statistical Learning and Intelligent Computation Hunan Changsha410205 China Applied Mathematics and Computational Science Program KAUST Thuwal23955-6900 Saudi Arabia Normandie Univ UNIHAVRE LMAH FR-CNRS-3335 ISCN Le Havre76600 France

The hierarchical small-world network is a real-world network. It models well the benefit transmission web of the pyramid selling in China and many other countries. In this paper, by applying the spectral graph theory, we study three important aspects of the consensus problem in the hierarchical small-world network: convergence speed, communication time-delay robustness, and network coherence. Firstly, we explicitly determine the Laplacian eigenvalues of the hierarchical small-world network by making use of its treelike structure. Secondly, we find that the consensus algorithm on the hierarchical small-world network converges faster than that on some well-studied sparse networks, but is less robust to time delay. The closed-form of the first-order and the second-order network coherence are also derived. Our result shows that the hierarchical small-world network has an optimal structure of noisy consensus dynamics. Therefore, we provide a positive answer to two open questions of Yi et al. Finally, we argue that some network structure characteristics, such as large maximum degree, small average path length, and large vertex and edge connectivity, are responsible for the strong robustness with respect to external perturbations. Copyright © 2023, The Authors. All rights reserved.

关键词： Eigenvalues and eigenfunctions

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