Showing papers on "Dirac delta function published in 2022"

Generalized Darmois–Israel Junction Conditions

Abstract: We present a general method to derive the appropriate Darmois–Israel junction conditions for gravitational theories with higher-order derivative terms by integrating the bulk equations of motion across the singular hypersurface. In higher-derivative theories, the field equations can contain terms which are more singular than the Dirac delta distribution. To handle them appropriately, we formulate a regularization procedure based on representing the delta function as the limit of a sequence of classical functions. This procedure involves imposing suitable constraints on the extrinsic curvature such that the field equations are compatible with the singular source being a delta distribution. As explicit examples of our approach, we demonstrate in detail how to obtain the generalized junction conditions for quadratic gravity, F(R) theories, a 4D low-energy effective action in string theory, and action terms that are Euler densities. Our results are novel, and refine the accuracy of previously claimed results in F(R) theories and quadratic gravity. In particular, when the coupling constants of quadratic gravity are those for the Gauss–Bonnet case, our junction conditions reduce to the known ones for the latter obtained independently by boundary variation of a surface term in the action. Finally, we briefly discuss a couple of applications to thin-shell wormholes and stellar models.

Direct calculation of the functional inverse of realistic interatomic potentials in field-theoretic simulations.

Abstract: We discuss the functional inverse problem in field-theoretic simulations for realistic pairwise potentials such as the Morse potential (widely used in particle simulations as an alternative to the 12-6 Lennard-Jones one), and we propose the following two solutions: (a) a numerical one based on direct inversion on a regular grid or deconvolution and (b) an analytical one by expressing attractive and repulsive contributions to the Morse potential as higher-order derivatives of the Dirac delta function; the resulting system of ordinary differential equations in the saddle-point approximation is solved numerically with appropriate model-consistent boundary conditions using a Newton-Raphson method. For the first time, exponential-like, physically realistic pair interactions are analytically treated and incorporated into a field-theoretic framework. The advantages and disadvantages of the two approaches are discussed in detail in connection with numerical findings from test simulations for the radial distribution function of a monatomic fluid at realistic densities providing direct evidence for the capability of the analytical method to resolve structural features down to the Angstrom scale.

Metaplectic representation and ordering (in)dependence in Vasiliev’s higher spin gravity

Abstract: We investigate the formulation of Vasiliev's four-dimensional higher-spin gravity in operator form, without making reference to one specific ordering. More precisely, we make use of the one-to-one mapping between operators and symbols thereof for a family of ordering prescriptions that interpolate between and go beyond Weyl and normal orderings. This correspondence allows us to perturbatively integrate the Vasiliev system in operator form and in a variety of gauges. Expanding the master fields in inhomogenous symplectic group elements, and letting products be controlled only by the group, we specify a family of factorized gauges in which we are able to integrate the system to all orders, producing exact solutions, including but not restricted to ones presented previously in the literature; and then connect, at first order, to a family of rotated Vasiliev gauges in which the solutions can be represented in terms of Fronsdal fields. The gauge function responsible for the latter transformation is explicitly constructed at first order. The analysis of the system in various orderings is facilitated by an analytic continuation of Gaussian symbols, by means of which one can distinguish and connect the two branches of the metaplectic double cover and give a rationale to the properties of the inner Klein operators as Gaussian delta sequences defining analytic delta densities. As an application of some of the techniques here developed, we evaluate twistor space Wilson line observables on our exact solutions and show their independence from auxiliary constructs up to the few first subleading orders in perturbation theory.

Benchmarks for Infinite Medium, Time Dependent Transport Problems with Isotropic Scattering

Abstract: Abstract The widely used AZURV1 transport benchmarks package provides a suite of solutions to isotropic scattering transport problems with a variety of initial conditions. Most of these solutions have an initial condition that is a Dirac delta function in space; as a result these benchmarks are challenging problems to use for verification tests in computer codes. Nevertheless, approximating a delta function in simulation often leads to low orders of convergence and the inability to test the convergence of high-order numerical methods. While there are examples in the literature of integration of these solutions as Green’s functions for the transport operator to produce results for more easily simulated sources, they are limited in scope and briefly explained. For a sampling of initial conditions and sources, we present solutions for the uncollided and collided scalar flux to facilitate accurate testing of source treatment in numerical solvers. The solution for the uncollided scalar flux is found in analytic form for some sources. Since integrating the Green’s functions is often nontrivial, discussion of integration difficulty and workarounds to find convergent integrals is included. Additionally, our uncollided solutions can be used as source terms in verification studies, in a similar way to the method of manufactured solutions.

Green's functions in quantum mechanics courses

Abstract: The use of Green's functions is valuable when solving problems in electrodynamics, solid-state physics, and many-body physics. However, its role in quantum mechanics is often limited to the context of scattering by a central force. This work shows how Green's functions can be used in other examples in quantum mechanics courses. In particular, we introduce time-independent Green's functions and the Dyson equation to solve problems with an external potential. We calculate the reflection and transmission coefficients of scattering by a Dirac delta barrier and the energy levels and local density of states of the infinite square well potential.

Strong solutions of impulsive pseudoparabolic equations

Abstract: We study the two-dimensional Cauchy problem for the quasilinear pseudoparabolic equation with a regular nonlinear minor term endowed with periodic initial data and periodicity conditions. The minor term depends on a small parameter ɛ>0 and, as ɛ→0, converges weakly⋆ to the expression incorporating the Dirac delta function, which models an instantaneous impulsive impact. We establish that the transition (shock) layer, associated with the Dirac delta function, is formed as ɛ→0, and that the family of strong solutions of the original problem converges to the strong solution of a two-scale microscopic–macroscopic model. This model consists of two equations and the set of initial and matching conditions, so that the ‘outer’ macroscopic solution beyond the transition layer is governed by the quasilinear homogeneous pseudoparabolic equation at the macroscopic (‘slow’) timescale, while the transition layer solution is defined at the microscopic level and obeys the semilinear pseudoparabolic equation at the microscopic (‘fast’) timescale. The latter is derived based on the microstructure of the transition layer profile.

Temperature-dependent dielectric function of intrinsic silicon: Analytic models and atom-surface potentials

Abstract: The optical properties of monocrystalline, intrinsic silicon are of interest for technological applications as well as fundamental studies of atom-surface interactions. For an enhanced understanding, it is of great interest to explore analytic models which are able to fit the experimentally determined dielectric function $\epsilon(T_\Delta, \omega)$, over a wide range of frequencies and a wide range of the temperature parameter $T_\Delta = (T-T_0)/T_0$, where $T_0 = 293\,{\rm K}$ represents room temperature. Here, we find that a convenient functional form for the fitting of the dielectric function of silicon involves a Lorentz-Dirac curve with a complex, frequency-dependent amplitude parameter, which describes radiation reaction. We apply this functional form to the expression $[\epsilon(T_\Delta, \omega) -1]/[ \epsilon(T_\Delta, \omega)+2]$, inspired by the Clausius-Mossotti relation. With a very limited set of fitting parameters, we are able to represent, to excellent accuracy, experimental data in the (angular) frequency range $0 < \omega < 0.16 \, {\rm a.u.}$ and $0< T_\Delta < 2.83$, corresponding to the temperature range $ 293\,{\rm K} < T < 1123\, {\rm K}$. Using our approach, we evaluate the short-range $C_3$ and the long-range $C_4$ coefficients for the interaction of helium atoms with the silicon surface. In order to validate our results, we compare to a separate temperature-dependent direct fit of $\epsilon(T_\Delta, \omega)$ to the Lorentz-Dirac model.

Band formation and defects in a finite periodic quantum potential

Abstract: Periodic quantum systems often exhibit energy spectra with well-defined energy bands separated by band gaps. The formation of band structure in periodic quantum systems is usually presented in the context of Bloch’s theorem or through other specialized techniques. Here we present a simple model of a finite one-dimensional periodic quantum system that can be used to explore the formation of band structure in a straightforward way. Our model consists of an infinite square well containing several evenly-spaced identical Dirac delta wells. Both attractive and repulsive delta wells are considered. We solve for the energy eigenvalues and eigenfunctions of this system directly and show the formation of band structure as the number of delta wells is increased, as well as how the size of the bands and gaps depends on the strength of the delta wells. These results are compared ot the predictions from Bloch’s theorem. In addition, we use this model to investigate how the energy spectrum is altered by the introduction of two types of defects in the periodicity of the system. Strength defects, in which the strength of one delta well is changed, can result in an energy level moving from one band, through the band gap, to another band as the strength of the well is varied. Position defects, in which the location of one delta well is changed, can modulate the size of the energy bands and sufficiently large position defects can move an energy level into a gap. Band structure and defects are important concepts for understanding many properties of quantum solids and this simple model provides an elementary introduction to these ideas.

Renormalized self-intersection local time for sub-bifractional Brownian motion

Abstract: Let SH,K = {SH,K(t), t ? 0} be a d?dimensional sub-bifractional Brownian motion with indices H ? (0, 1) and K ? (0,1]. Assuming d ? 2, as HKd < 1, we mainly prove that the renormalized self-intersection local time ? t0 ? s0 ?(SH,K(s) ? SH,K(r))drds ? E [?t0 ?s0 ?(SH,K(s) ? SH,K(r))drds] exists in L2, where ?(x) is the Dirac delta function for x ? Rd.

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Abstract: In this note the two dimensional Dirac operator Aη with an electrostatic δ -shell interaction of strength η∈R supported on a straight line is studied. We observe a spectral transition in the sense that for the critical interaction strengths η=±2 the continuous spectrum of Aη inside the spectral gap of the free Dirac operator A0 collapses abruptly to a single point.

On Aspects of Gradient Elasticity: Green's Functions and Concentrated Forces

Abstract: In the first part of our review paper, we consider the problem of approximating the Green’s function of the Lagrange chain by continuous analogs. It is shown that the use of continuous equations based on the two-point Padé approximants gives good results. In the second part of the paper, the problem of singularities arising in the classical theory of elasticity with affecting concentrated loadings is considered. To overcome this problem, instead of a transition to the gradient theory of elasticity, it is proposed to change the concept of concentrated effort. Namely, the Dirac delta function is replaced by the Whittaker–Shannon–Kotel’nikov interpolating function. The only additional parameter that characterizes the microheterogeneity of the medium is used. An analog of the Flamant problem is considered as an example. The found solution does not contain singularities and tends to the classical one when the microheterogeneity parameter approaches zero. The derived formulas have a simpler form compared to those obtained by the gradient theory of elasticity.

Bound states of a one-dimensional Dirac equation with multiple delta-potentials

Abstract: Two approaches are developed for the study of the bound states of a one-dimensional Dirac equation with the potential consisting of N δ-function centers. One of these uses Green’s function method. This method is applicable to a finite number N of δ-point centers, reducing the bound state problem to finding the energy eigenvalues from the determinant of a 2 N × 2 N matrix. The second approach starts with the matrix for a single delta-center that connects the two-sided boundary conditions for this center. This connection matrix is obtained from the squeezing limit of a piecewise constant approximation of the delta-function. Having then the connection matrices for each center, the transmission matrix for the whole system is obtained by multiplying the one-center connection matrices and the free transfer matrices between neighbor centers. An equation for bound state energies is derived in terms of the elements of the total transfer matrix. Within both approaches, the transcendental equations for bound state energies are derived, the solutions to which depend on the strength of delta-centers and the distance between them, and this dependence is illustrated by numerical calculations. The bound state energies for the potentials composed of one, two, and three delta-centers ( N = 1, 2, 3) are computed explicitly. The principle of strength additivity is analyzed in the limits as the delta-centers merge at a single point or diverge to infinity.

3E-Solver: An Effortless, Easy-to-Update, and End-to-End Solver with Semi-Supervised Learning for Breaking Text-Based Captchas

Abstract: Text-based captchas are the most widely used security mechanism currently. Due to the limitations and specificity of the segmentation algorithm, the early segmentation-based attack method has been unable to deal with the current captchas with newly introduced security features (e.g., occluding lines and overlapping). Recently, some works have designed captcha solvers based on deep learning methods with powerful feature extraction capabilities, which have greater generality and higher accuracy. However, these works still suffer from two main intrinsic limitations: (1) many labor costs are required to label the training data, and (2) the solver cannot be updated with unlabeled data to recognize captchas more accurately. In this paper, we present a novel solver using improved FixMatch for semi-supervised captcha recognition to tackle these problems. Specifically, we first build an end-to-end baseline model to effectively break text-based captchas by leveraging encoder-decoder architecture and attention mechanism. Then we construct our solver with a few labeled samples and many unlabeled samples by improved FixMatch, which introduces teacher forcing, adaptive batch normalization, and consistency loss to achieve more effective training. Experiment results show that our solver outperforms state-of-the-arts by a large margin on current captcha schemes. We hope that our work can help security experts to revisit the design and usability of text-based captchas. The source code of this work is available at https://github.com/SJTU-dxw/3E-Solver-CAPTCHA.

Thermal analysis for clamped laminated beams with non-uniform temperature boundary conditions

Abstract: The focus of this study is thermal responses of a clamped composite laminated beam with arbitrary layer numbers under non-uniform temperature boundary conditions. Analytical solutions of temperature, stresses, and displacements are derived based on the theory of thermoelasticity. The temperature distribution in the laminated beam is divided into two parts. The first part is constructed to satisfy the inhomogeneous temperature boundary conditions, while the second part is obtained on the basis of the Fourier law of heat conduction and the temperature environment. On the other hand, the unit pulse function and the Dirac delta function are introduced to translate the clamped support into the simply support with an unknown horizontal stress. According to the continuities at the interface and the state space method, the relationships of displacements and stresses between the top and bottom surfaces of the laminated beam are derived. Finally, the unknown coefficients of displacements and stresses are determined by the mechanical boundary conditions. It can be observed from the numerical results that this method has excellent convergence performance. The accuracy of this method is verified by comparing with the results of the finite element method. Furthermore, the effects of surface temperature, material properties, length-to-thickness ratio and layer numbers on the distributions of temperature, displacements and stresses in the laminated beam are in depth investigated.

A Universal PINNs Method for Solving Partial Differential Equations with a Point Source

Abstract: In recent years, deep learning technology has been used to solve partial differential equations (PDEs), among which the physics-informed neural networks (PINNs)method emerges to be a promising method for solving both forward and inverse PDE problems. PDEs with a point source that is expressed as a Dirac delta function in the governing equations are mathematical models of many physical processes. However, they cannot be solved directly by conventional PINNs method due to the singularity brought by the Dirac delta function. In this paper, we propose a universal solution to tackle this problem by proposing three novel techniques. Firstly the Dirac delta function is modeled as a continuous probability density function to eliminate the singularity at the point source; secondly a lower bound constrained uncertainty weighting algorithm is proposed to balance the physics-informed loss terms of point source area and the remaining areas; and thirdly a multi-scale deep neural network with periodic activation function is used to improve the accuracy and convergence speed. We evaluate the proposed method with three representative PDEs, and the experimental results show that our method outperforms existing deep learning based methods with respect to the accuracy, the efficiency and the versatility.

Discontinuous Galerkin method for linear wave equations involving derivatives of the Dirac delta distribution

Abstract: Linear wave equations sourced by a Dirac delta distribution $\delta(x)$ and its derivative(s) can serve as a model for many different phenomena. We describe a discontinuous Galerkin (DG) method to numerically solve such equations with source terms proportional to $\partial^n \delta /\partial x^n$. Despite the presence of singular source terms, which imply discontinuous or potentially singular solutions, our DG method achieves global spectral accuracy even at the source's location. Our DG method is developed for the wave equation written in fully first-order form. The first-order reduction is carried out using a distributional auxiliary variable that removes some of the source term's singular behavior. While this is helpful numerically, it gives rise to a distributional constraint. We show that a time-independent spurious solution can develop if the initial constraint violation is proportional to $\delta(x)$. Numerical experiments verify this behavior and our scheme's convergence properties by comparing against exact solutions.

Computational Reconstruction of Dirac- Delta Function from Quantum Electrodynamics Rules

Abstract: It is reconstructed the Dirac-Delta function exhibiting an interesting texture when the function is parametrized. For this end the well-known quantum electrodynamics rules are employed. Thus when the energy conservation is done then a closed form can be extracted and that is far away from its traditional meaning.