High-order Expansion Research Articles

Kitaev honeycomb antiferromagnet in a field: quantum phase diagram for general spin

We use tensor-network methods and high-order linked-cluster expansions to explore the quantum phase diagram of the antiferromagnetic Kitaev honeycomb model in a magnetic field for general spin S values. Tensor network calculations for the pure Kitaev model confirm the absence of fluxes and spin-spin correlations beyond nearest neighbors, while revealing discrete orientational symmetry breaking for S ∈ 1, 3/2, 2, consistent with the semiclassical limit. An intermediate region between Kitaev phases and the high-field polarized phase is identified for all considered spin values, showing a sequence of potential phases characterized by distinct local magnetization patterns while the total magnetization increases smoothly as a function of the field. Linked-cluster expansions for the high-field zero-momentum gap and spectral weight indicate a quantum critical breakdown of the polarized phase, suggesting exotic physics at intermediate Kitaev couplings.

Fine large-time asymptotics for the axisymmetric Navier–Stokes equations

We examine the large-time behavior of axisymmetric solutions without swirl of the Navier–Stokes equation in R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^3$$\\end{document}. We construct higher-order asymptotic expansions for the corresponding vorticity. The appeal of this work lies in the simplicity of the applied techniques: Our approach is completely based on standard L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document}-based entropy methods.

Asymptotic profiles of zero points of solutions to the heat equation

In this paper, we consider the asymptotic profiles of zero points for the spatial variable of the solutions to the heat equation. By giving suitable conditions for the initial data, we prove the existence of zero points by extending the high-order asymptotic expansion theory for the heat equation. This reveals a previously unknown asymptotic profile of zero points diverging at O ( t ) O(t) . In a one-dimensional spatial case, we show the zero point’s second and third-order asymptotic profiles in a general situation. We also analyze a zero level set in high-dimensional spaces and obtain results that extend the results for the one-dimensional spatial case.

Kinematics of nonlinear waves over variable bathymetry. Part I: Numerical modelling, verification and validation

Fluid particle kinematics due to wave motion (i.e. orbital velocities and accelerations) at and beneath the free surface is involved in many coastal and ocean engineering applications, e.g. estimation of wave-induced forces on structures, sediment transport, etc. This work presents the formulations of these kinematics fields within a fully nonlinear potential flow (FNPF) approach. In this model, the velocity potential is approximated with a high-order polynomial expansion over the water column using an orthogonal basis of Chebyshev polynomials of the first kind. Using the same basis, original analytical expressions of the components of velocity and acceleration are derived in this work. The estimation of particle accelerations in the course of the simulation involves the time derivatives of the decomposition coefficients, which are computed with a high-order backward finite-difference scheme in time. The capability of the numerical model in computing the particle kinematics is first validated for regular nonlinear waves propagating over a flat bottom. The model is shown to be able to predict both the velocity and acceleration of highly nonlinear and nearly breaking waves with negligible error compared to the corresponding stream function wave solution. Then, for regular waves propagating over an uneven bottom (bar-type bottom profile), the simulated results are confronted with existing experimental data, and very good agreement is achieved up to the sixth-order harmonics for free surface elevation, velocity and acceleration.

Exploring Weak Gravitational Lensing Effects by a Regular Black Hole

Gravitational lensing is one of the physical consequences of Einstein’s general theory of relativity, which has been observationally confirmed. This work aims to study the weak gravitational lensing scenario where the lens is a spherically symmetric, charged-nonsingular black hole that asymptotically behaves as the Reissner-Nordström (RN). The difference exists in the higher-order expansion. Therefore, the metric function of the regular charged black hole is expanded up to the fourth order, and the deflection angle is calculated. The thin lens equation is used, leading to five images: two imaginary and three real images. The exact positions of the images and magnification properties are calculated by providing the physical parameters such as mass, charge, and distances. Our calculation shows that the third image position shifted away from the optical axis. The formalism is applied to the case of a supermassive black hole located at the center of our galaxy with the assumption that it has an electric charge.

Path-dependent correlations in dynamically tuned Ising models and its short-time behavior: Application of Magnus expansion

We study both analytically and numerically the buildup of antiferromagnetic (AF) correlation in the dynamically tuned Ising models. In short-time scale, we apply Magnus expansion (ME) to derive the high-order analytic expression of the connected correlation functions and compare it with exactly numerical results for the different lattice geometries, e.g., 1D chain, 2×n lattice, and n×n lattice. It is shown that the high-order expansion is required to describe accurately the buildup of AF correlation in the quench dynamics. Moreover, through a 2D square lattice, we find that the magnitude of AF correlation for the same Manhattan distance is proportional to the number of the shortest paths in a sufficiently long time until long and distinct paths are involved significantly. Finally, we propose an applicable experimental setup to realize our findings.

The Kinematic Models of the SINS and Its Errors on the SE(3) Group in the Earth-Centered Inertial Coordinate System.

In this paper, the kinematic models of the Strapdown Inertial Navigation System (SINS) and its errors on the SE(3) group in the Earth-Centered Inertial frame (ECI) are established. On the one hand, with the ECI frame being regarded as the reference, based on the joint representation of attitude and velocity on the SE(3) group, the dynamic of the local geographic coordinate system (n-frame) and the body coordinate system (b-frame) evolve on the differentiable manifold, respectively, and the high-order expansion of the Baker-Campbell-Haussdorff equation compensates for the non-commutative motion errors stimulated by strong maneuverability. On the other hand, the kinematics of the left- and right-invariant errors of the n-frame and the b-frame on the SE(3) group are separately derived, where the errors of the b-frame completely depend on inertial sensor errors, while the errors of the n-frame rely on position errors and velocity errors. In this way, the errors brought by the inconsistency of the reference coordinate system are tackled, and a novel attitude error definition is introduced to separate and decouple the factors affecting the dynamic of the n-frame errors and the b-frame errors for better attitude estimation. Through a turntable experiment and a car-mounted field experiment, the effectiveness of the proposed kinematic models in estimating attitude has been verified, with a remarkable improvement in yaw angle accuracy in the case of large initial misalignment angles, and the models developed have better robustness compared to the traditional SE(3) group-based model.

Monte Carlo Based Techniques for Quantum Magnets with Long-Range Interactions.

Long-range interactions are relevant for a large variety of quantum systems in quantum optics and condensed matter physics. In particular, the control of quantum-optical platforms promises to gain deep insights into quantum-critical properties induced by the long-range nature of interactions. From a theoretical perspective, long-range interactions are notoriously complicated to treat. Here, we give an overview of recent advancements to investigate quantum magnets with long-range interactions focusing on two techniques based on Monte Carlo integration. First, the method of perturbative continuous unitary transformations where classical Monte Carlo integration is applied within the embedding scheme of white graphs. This linked-cluster expansion allows extracting high-order series expansions of energies and observables in the thermodynamic limit. Second, stochastic series expansion quantum Monte Carlo integration enables calculations on large finite systems. Finite-size scaling can then be used to determine the physical properties of the infinite system. In recent years, both techniques have been applied successfully to one- and two-dimensional quantum magnets involving long-range Ising, XY, and Heisenberg interactions on various bipartite and non-bipartite lattices. Here, we summarise the obtained quantum-critical properties including critical exponents for all these systems in a coherent way. Further, we review how long-range interactions are used to study quantum phase transitions above the upper critical dimension and the scaling techniques to extract these quantum critical properties from the numerical calculations.

Combining the Asymptotic Numerical Method with the Harmonic Balance Method to capture the nonlinear dynamics of spur gears

The present work proposes the Asymptotic Numerical Method (ANM) combined to the Harmonic Balance Method (HBM) as a valuable approach to solve the nonlinear dynamics of gear pairs. The ANM is a continuation method based on high-order Taylor series expansion of the computed solution branch. The HBM is a periodic solution representation method based on high-order Fourier series. Thanks to a quadratic recast of the equation of motion, the Taylor and Fourier series can be computed in a very efficient way and each step produces a continuous representation of the solution branch making the continuation very robust. By employing this method, the periodic solutions may be easily expressed with respect to both the shaft rotation frequency and the gear mesh frequency as the adoption of a high number of harmonics has negligible effects on the computational burden. Effectiveness and reliability of the method are proven by comparing the numerical results with that obtained from the Runge–Kutta time integration scheme and experimental data from literature. Afterwards, a comparison in terms of computational efficiency is performed. Finally, some considerations are drawn in order to highlights the main differences between the two methods within gear dynamics computation.

Connecting microscopic and mesoscopic mechanics in model structural glasses.

We present a novel formalism to characterize elastic heterogeneities in amorphous solids. In particular, we derive high-order strain-energy expansions for pairwise energies under athermal quasistatic dynamics. We then use the presented formalism to study the statistical properties of pairwise expansion coefficients and their link with the statistics of soft, quasilocalized modes, for a wide range of formation histories in both two- and three-dimensional systems. We further exploit the presented framework to access local yield stress maps by performing a non-linear stress-strain expansion within a cavity embedded in a frozen matrix. We show that our "bond micromechanics" compare well with the original "frozen matrix" method, with the caveat of overestimating large stress activations. We additionally show how local yield rules can be used as an input for a scalar elasto-plastic model (EPM) to predict the stress response of materials ranging from ductile to brittle. Finally, we highlight some of the limits of simple mesoscale models in capturing the aging dynamics of post-yielding systems. Intriguingly, we observe subdiffusive and diffusive shear band growths for particle-based simulations and EPMs, respectively.

Polynomial chaos expansions for modelling the statistics of acoustic propagation in random waveguides

The use of Polynomial Chaos expansions to model the transfer of the statistical properties of the sound speed in ocean waveguides to those of acoustic waves passing through them is described. A perturbational framework approximating the interaction of acoustic normal modes with fluctuations around a background sound speed is used, with the fluctuations being represented vertically by empirical orthogonal functions and horizontally by correlation functions. The Polynomial Chaos expansions are derived to second order in the uncorrelated random variable representing the sound speed fluctuations for the generalized phase of the complex modal amplitudes, and to third order for the modal amplitudes themselves. The ability of the second order polynomial chaos expansion for the generalized model phase to accurately model acoustic propagation statistics is closely coupled to the accuracy of the adiabatic approximation in light scattering regimes. The weights of the Polynomial Chaos expansions are obtained using both direct (theoretical) and indirect (least-squares fit) methods. Divergence between these two sets of weights increases with combinations of increasing strength of sound speed fluctuations, increasing frequency, or increasing range, indicating where a higher order expansion may be necessary. Modelling results for the first and second moments of the acoustic intensity and the Scintillation Index in random waveguides are presented and compared to Monte Carlo results.

The effects of general relativity on close-in radial-velocity-detected exosystems

Aims. The detection of the first exoplanet around a solar-type star revealed the existence of close-in planets. Several of these close-in planets are part of multi-planet systems. For systems detected via the radial velocity (RV) method, we lack information on the mutual inclination of the orbital planes. The aim of this work is to study the long-term stability of RV-detected two-planet systems with close-in planets and identify possible three-dimensional configurations for these systems that are compatible with observations. To do so, we focused on the protective mechanism of the Lidov-Kozai (LK) secular resonance and studied the effects of general relativity (GR) on long-term evolution. Methods. By means of an analytical study based on a high-order secular Hamiltonian expansion in the eccentricities and inclinations, we first identified ranges of values for the orbital and mutual inclinations that are compatible with the presence of the LK resonance in the purely gravitational case. Then, adding the secular contribution of the relativistic corrections exerted by the central star on the inner planet, namely the advance of its pericenter precession, we analysed the outcomes of the two sets of simulations. We compared our results to analytical estimates to determine the importance of GR effects. Results. We find that for the majority of the systems considered, GR strongly affects the dynamics of the system and, most of the time, voids the LK resonance, as observed for GJ 649, GJ 832, HD 187123, HD 190360, HD 217107, and HD 47186. The long-term stability of these systems is then possible whatever the mutual inclination of the orbits. On the contrary, for GJ 682, HD 11964, HD 147018, and HD 9446, the LK resonant region in the parameter space of the orbital and mutual inclinations is left (almost) unchanged when GR effects are considered, and consequently their long-term stability is only possible if the mutual inclination of the orbits is low or if the systems are in the LK regime with a high mutual inclination.

Higher-order expansions of powered extremes of logarithmic general error distribution

Higher-order expansions of powered extremes of logarithmic general error distribution

Transient Thermal Analysis of Space Rectangular Thin-Walled Structures by Carrera Unified Formulation

The accurate prediction of transient thermal fields of space structures is of paramount significance for the analysis of geometrical stability and thermal management systems. This paper proposes a novel method to solve the transient thermal field of the thin-walled cross section of beams in outer space. High-order beam elements are derived by means of the Carrera unified formulation (CUF), where the longitudinal direction of the beam is discretized by one-dimensional two-node and four-node elements, whereas arbitrary high-order expansion functions are used for the description of the temperature expansion within the cross section. The governing differential equations of the transient temperature field are derived according to the transient thermal conduction theory and the weighted residual method. The solution of the initial value problem for ordinary differential equations is obtained through the application of the Adams–Bashforth method, leading to the determination of the transient thermal field. In order to demonstrate the accuracy of the temperature results, a convergence analysis is conducted. Finally, temperature distributions within the cross section and along the longitudinal direction of the model are discussed to show the effects of several factors, including material properties, angle of solar radiation, and shadows induced by the shelter of other components.

On the Importance of the Recovery Procedure in the Semi-Analytical Solution for the Static Analysis of Curved Laminated Panels: Comparison with 3D Finite Elements.

The manuscript presents an efficient semi-analytical solution with three-dimensional capabilities for the evaluation of the static response of laminated curved structures subjected to general external loads. A two-dimensional model is presented based on the Equivalent Single Layer (ESL) approach, where the displacement field components are described with a generalized formulation based on a higher-order expansion along the thickness direction. The fundamental equations are derived from the Hamiltonian principle, and the solution is found by means of Navier's approach. Then, an efficient recovery procedure, derived from the three-dimensional elasticity equations and based on the Generalized Differential Quadrature (GDQ) method, is adopted for the derivation of the three-dimensional solution. Some examples of investigation are presented, where the numerical predictions of refined three-dimensional Finite-Element-based models are matched with a high level of accuracy. The model is validated for both straight and curved panels, taking into account different lamination schemes and load shapes. Furthermore, it is shown that the numerical solution to the elasticity problem in the recovery procedure is determining and accurately predicting the three-dimensional static response of the doubly-curved shell solid.

A positivity-preserving, linear, energy stable and convergent numerical scheme for the Poisson–Nernst–Planck (PNP) system

This article focuses on the convergence analysis for a fully discrete finite difference scheme for the time-dependent Poisson–Nernst–Planck system. The numerical scheme, a three-level linearized finite difference algorithm based on the reformulation of the Nernst–Planck equations, which was developed in [J. Sci. Comput. 81(2019),436–458]. The positivity-preserving property and energy stability were theoretically established. In this paper, we rigorously prove first-order convergence in time and second-order convergence in space for the numerical scheme in the ℓ∞(0,T;ℓ2)∩ℓ2(0,T;Hh1) norm under the condition that time step size is linearly proportional to spatial mesh size, in which the natural property of the exponential terms gives a lot of challenges. Moreover, the higher-order asymptotic expansion (up to third-order temporal accuracy and fourth-order spatial accuracy) has to be involved, due to the leading local truncation error will not be enough to recover an ℓ∞ bound for ion concentrations n and p. To our knowledge, this scheme will be the first linear decoupled algorithm to combine the following theoretical properties for the PNP system: ion concentration positivity preserving, unconditionally energy stability and optimal rate convergence. Numerical results are shown to be consistent with theoretical analysis.

HOPE: High-Order Polynomial Expansion of Black-Box Neural Networks.

Despite their remarkable performance, deep neural networks remain mostly "black boxes", suggesting inexplicability and hindering their wide applications in fields requiring making rational decisions. Here we introduce HOPE (High-order Polynomial Expansion), a method for expanding a network into a high-order Taylor polynomial on a reference input. Specifically, we derive the high-order derivative rule for composite functions and extend the rule to neural networks to obtain their high-order derivatives quickly and accurately. From these derivatives, we can then derive the Taylor polynomial of the neural network, which provides an explicit expression of the network's local interpretations. We combine the Taylor polynomials obtained under different reference inputs to obtain the global interpretation of the neural network. Numerical analysis confirms the high accuracy, low computational complexity, and good convergence of the proposed method. Moreover, we demonstrate HOPE's wide applications built on deep learning, including function discovery, fast inference, and feature selection. We compared HOPE with other XAI methods and demonstrated our advantages. The code is available at https://github.com/HarryPotterXTX/HOPE.git.

High-order models for convection–diffusion-reaction transport in multiscale porous media

Developing highly accurate models for predicting the convection–diffusion-reaction (CDR) transport in hierarchical porous media with strong heterogeneities on multiple scales is crucial but not yet available. In this work, an innovative high-order multiscale computational framework is developed to capture the local and global variation characteristics of flow fields and reactant concentration at multiple scales. The homogenized solutions and macro-meso high-order solutions are established by the formal two-scale asymptotic analysis. By directly expanding the mesoscopic cell functions to the microscopic levels, the three-scale high-order models are built by assembling the meso-micro high-order expansions of mesoscopic cell functions and macro-meso low/high-order models. The present approaches follow the reverse thought process of the reiteration homogenization method, and provide a very innovative way to develop highly accurate and efficient solutions for the CDR coupling problems in multiscale porous media. The effectiveness and accuracy of the proposed multiscale models are validated by several representative cases.

Scalar QNM spectra of Kerr and Reissner-Nordström revealed by eigenvalue repulsions in Kerr-Newman

Recent studies of the gravito-electromagnetic frequency spectra of Kerr-Newman (KN) black holes have revealed two families of quasinormal modes (QNMs), namely photon sphere modes and near-horizon modes. However, they can only be unambiguously distinguished in the Reissner-Nordström (RN) limit, due to a phenomenon called eigenvalue repulsion (also known as level repulsion, avoided crossing or the Wigner-Teller effect), whereby the two families can interact strongly near extremality. We find that these features are also present in the QNM spectra of a scalar field in KN, where the perturbation modes are described by ODEs and thus easier to explore. Starting from the RN limit, we study how the scalar QNM spectra of KN dramatically changes as we vary the ratio of charge to angular momentum, all the way until the Kerr limit, while staying at a fixed distance from extremality. This scalar field case clarifies the (so far puzzling) relationship between the QNM spectra of RN and Kerr black holes and the nature of the eigenvalue repulsions in KN, that ultimately settle the fate of the QNM spectra in Kerr. We study not just the slowest-decaying QNMs (both for ℓ = m = 0 and ℓ = m = 2), but several sub-dominant overtones as well, as these turn out to play a crucial role understanding the KN QNM spectra. We also give a new high-order WKB expansion of KN QNMs that typically describes the photon sphere modes beyond the eikonal limit, and use a matched asymptotic expansion to get a very good approximation of the near-horizon modes near extremality.

Asymptotic analysis of the narrow escape problem in general shaped domain with several absorbing necks

This paper considers the two-dimensional narrow escape problem in a domain which is composed of a relatively big head and several absorbing narrow necks. The narrow escape problem is to compute the mean first passage time (MFPT) of a Brownian particle traveling from inside the head to the end of the necks. The original model for MFPT is to solve a mixed Dirichlet–Neumann boundary value problem for the Poisson equation in the composite domain, and is computationally challenging. In this paper, we compute the MFPT by solving an equivalent Neumann–Robin type boundary value problem. By solving the new model, we obtain the three-term asymptotic expansion of the MFPT. We also conduct numerical experiments to show the accuracy of the high order expansion. As far as we know, this is the first result on high order asymptotic solution for narrow escape problem (NEP) in a general shaped domain with several absorbing neck windows. This work is motivated by Li (2014 J. Phys. A: Math. Theor. 47 505202), where the Neumann–Robin model was proposed to solve the NEP in a domain with a single absorbing neck.