Dirac Δ-function Research Articles

Structure of pressure-gradient-driven current singularity in ideal magnetohydrodynamic equilibrium

Singular currents typically appear on rational surfaces in non-axisymmetric ideal magnetohydrodynamic (MHD) equilibria with a continuum of nested flux surfaces and a continuous rotational transition. These currents have two components: a surface current (Dirac δ-function in flux surface labeling) that prevents the formation of magnetic islands, and an algebraically divergent Pfirsch–Schlüter current density when a pressure gradient is present across the rational surface. On flux surfaces adjacent to the rational surface, the traditional treatment gives the Pfirsch–Schlüter current density scaling as , where is the difference of the rotational transform relative to the rational surface. If the distance s between flux surfaces is proportional to , the scaling relation will lead to a paradox that the Pfirsch–Schlüter current is not integrable. In this work, we investigate this issue by considering the pressure-gradient-driven singular current in the Hahm–Kulsrud–Taylor problem, which is a prototype for singular currents arising from resonant magnetic perturbations. We show that not only the Pfirsch–Schlüter current density but also the diamagnetic current density are divergent as . However, due to the formation of a Dirac δ-function current sheet at the rational surface, the neighboring flux surfaces are strongly packed with . Consequently, the singular current density , making the total current finite, thus resolving the paradox. Furthermore, the strong packing of flux surfaces causes a steepening of the pressure gradient near the rational surface, with . In general non-axisymmetric MHD equilibrium, contrary to Grad’s conjecture that the pressure profile is flat around densely distributed rational surfaces, our result suggests a pressure profile that densely steepens around them.

A hybridizable discontinuous Galerkin method on unfitted meshes for single-phase Darcy flow in fractured porous media

We present a novel hybridizable discontinuous Galerkin (HDG) method on unfitted meshes for single-phase Darcy flow in a fractured porous medium. In particular we apply the HDG methodology to the recently introduced reinterpreted discrete fracture model (RDFM) that use Dirac-δ functions to model both conductive and blocking fractures. Due to the use of Dirac-δ function approach for the fractures, our numerical scheme naturally allows for unfitted meshes with respect to the fractures, which is the major novelty of the proposed scheme. Moreover, the scheme is locally mass conservative. In particular, our scheme has a simple form, which is a novel modification of an existing regular Darcy flow HDG solver by adding the following two components: (i) locate the co-dimension one fractures in the background mesh and add the appropriate surface integrals associated with these fractures into the stiffness matrix, (ii) adjust the penalty parameters on cells cut through conductive and blocking fractures (fractured cells).Despite the simplicity of the proposed scheme, it performs quite well for various benchmark test cases in both two and three dimensions. To the best of our knowledge, this is the first time that an unfitted finite element scheme been applied to complex fractured porous media flow problems in 3D with both blocking and conductive fractures without any restrictions on the meshes.

Determination of the green function of a pulsed acoustic source in a uniform homogeneous flow with an arbitrary Mach number

The wave field created by a pulsed point source of sound in a uniform homogeneous flow with an arbitrary value of the Mach number is theoretically studied. The aim of research is to obtain an analytical dependence of the sound field on physical parameters.
 The space-waveguide Fourier expansion of the sound field is used to solve the Cauchy problem for the wave equation in a reference frame moving together with the medium. It is only necessary to transform the spatiotemporal dependence of the source, given in a fixed frame of reference, to a dependence in a moving frame of reference.
 The transition to the description of the solution in the frame of reference, relative to which the medium moves at a constant velocity, is made taking into account the main properties of the generalized Dirac δ-function.
 Analytical dependences of the sound field on physical parameters are obtained for both subsonic and supersonic flows. A comparison is made with the results of calculations for the case of a pulsed point source moving in a medium at rest. The solution obtained in this work for the case of an impulsive source moving in a medium at rest made it possible to trace its connection with the Green's function for a stationary source in a moving medium. The analytical dependence of the obtained solution for the Green's function makes it possible to write down the explicit form of the "characteristic solution surface", that is, the "wave front". It is shown that the difference between the solutions for subsonic and supersonic motion is characterized by the position of the source relative to the moving wavefront.
 The calculation results can be used to describe the sound field created by an arbitrary spatiotemporal distribution of sound sources

Variation of Gini and Kolkata indices with saving propensity in the Kinetic Exchange model of wealth distribution: An analytical study

We study analytically the change in the wealth (x) distribution P(x) against saving propensity λ in a closed economy, using the Kinetic theory. We estimate the Gini (g) and Kolkata (k) indices by deriving (using P(x)) the Lorenz function L(f), giving the cumulative fraction L of wealth possessed by fraction f of the people ordered in ascending order of wealth. First, using the exact result for P(x) when λ=0 we derive L(f), and from there the index values g and k. We then proceed with an approximate gamma distribution form of P(x) for non-zero values of λ. Then we derive the results for g and k at λ=0.25 and as λ→1. We note that for λ→1 the wealth distribution P(x) becomes a Dirac δ-function. Using this and assuming that form for larger values of λ we proceed for an approximate estimate for P(x) centered around the most probable wealth (a function of λ). We utilize this approximate form to evaluate L(f), and using this along with the known analytical expression for g, we derive an analytical expression for k(λ). These analytical results for g and k at different λ are compared with numerical (Monte Carlo) results from the study of the Chakraborti–Chakrabarti model. Next we derive analytically a relation between g and k. From the analytical expressions of g and k, we proceed for a thermodynamic mapping to show that the former corresponds to entropy and the latter corresponds to the inverse temperature.

A hybrid-mixed finite element method for single-phase Darcy flow in fractured porous media

We present a hybrid-mixed finite element method for a novel hybrid-dimensional model of single-phase Darcy flow in a fractured porous media. In this model, the fracture is treated as a (d−1)-dimensional interface within the d-dimensional fractured porous domain, for d=2,3. Two classes of fracture are distinguished based on the permeability magnitude ratio between the fracture and its surrounding medium: when the permeability in the fracture is (significantly) larger than in its surrounding medium, it is considered as a conductive fracture; when the permeability in the fracture is (significantly) smaller than in its surrounding medium, it is considered as a blocking fracture. The conductive fractures are treated using the classical hybrid-dimensional approach of the interface model where pressure is assumed to be continuous across the fracture interfaces, while the blocking fractures are treated using the recent Dirac-δ function approach where normal component of Darcy velocity is assumed to be continuous across the interface. Due to the use of Dirac-δ function approach for the blocking fractures, our numerical scheme allows for nonconforming meshes with respect to the blocking fractures. This is the major novelty of our model and numerical discretization. Moreover, our numerical scheme produces locally conservative velocity approximations and leads to a symmetric positive definite linear system involving pressure degrees of freedom on the mesh skeleton only. As an application, we extend the idea to a simple transport model. The performance of the proposed method is demonstrated by various benchmark test cases in both two- and three-dimensions. Numerical results indicate that the proposed scheme is highly competitive with existing methods in the literature

2D Schrödinger operators with singular potentials concentrated near curves

ABSTRACT We investigate the Schrödinger operators in with the short-range potentials which are localized around a smooth closed curve γ. The operators can be viewed as an approximation of the heuristic Hamiltonian , where is Dirac's δ-function supported on γ and is its normal derivative on γ. Assuming that the operator has only discrete spectrum, we analyse the asymptotic behaviour of eigenvalues and eigenfunctions of . The transmission conditions on γ for the eigenfunctions , which arise in the limit as reveal a nontrivial connection between spectral properties of and the geometry of γ.

Asymmetric Perfect Absorption and Lasing of Nonlinear Waves by a Complex δ-Potential

Spectral singularities and coherent perfect absorption are two interrelated concepts that have originally been introduced and studied for linear waves interacting with complex potentials. In the meantime, the distinctive asymptotic behavior of perfectly absorbed waves suggests considering possible generalizations of these phenomena for nonlinear waves. Here, we address the perfect absorption of nonlinear waves by an idealized infinitely narrow dissipative potential modeled by a Dirac δ-function with an imaginary amplitude. Our main result is the existence of perfectly absorbed flows whose spatial amplitude distributions are asymmetric with respect to the position of the absorber. These asymmetric states do not have a linear counterpart. Their linear stability is verified numerically. The nonlinear waveguide also supports symmetric and constant-amplitude perfectly absorbed flows. The stability of solutions of the latter type can be confirmed analytically.

Calculated Study of the Influence of Overheating Aluminum Melt on the Dynamics the Granulation Process

A three-dimensional mathematical model of the solidification process of a liquid metal is considered, taking into account the mobility of the boundaries at which the phase transition is carried out (Stefan boundary value problem). The algorithm of calculation is improved, allowing due to the use of the Dirac δ-function in determining the effective heat capacity to take into account the nonlinearity of the equation of unsteady thermal conductivity and the heat of the phase transition. A numerical study of heat transfer during solidification of lead-containing aluminum melt droplets in air and water is carried out. The influence of droplet size and melt overheating on the solidification dynamics of granules has been studied. An approximate ratio based on the square root law is proposed, taking into account the amount of overheating of the liquid phase and linking the thickness of the formed solid phase with the duration of the granulation process

Green's function formulation of multiple nonlinear Dirac δ-function potential in one dimension

In this work, we study the scattering problem of the general nonlinear finitely many Dirac delta potentials with complex coupling constants (or opacities in the context of optics) using the Green's function method and then find the bound state energies and the wave functions for the particular form of the nonlinearity in the case of positive real coupling constants.

Grafted optical vortex with controllable orbital angular momentum distribution.

In an optical vortex (OV) field, the orbital angular momentum (OAM) distribution strongly depends on the intensity, which results in difficulty in OAM independent modulation. To overcome this limitation, we propose a grafted optical vortex (GOV) via spiral phase reconstruction of two or more OVs with different topological charges (TCs). To remain the annular shape of the GOV's intensity, the Dirac δ-function is employed to restrict the energy in a ring. Theoretical analysis and manipulation experiments of polystyrene microspheres show that the magnitude and direction of the GOV's local OAM are controllable by modulating the grafted TCs while the intensity remains constant. The results of this work provide an ingenious method to control the local tangential force on the light ring, which will promote potential applications in optical trapping and rotating micro-particles.

Fluxon interaction with the finite-size dipole impurity

Interaction of the fluxon with the finite size dipole impurity in the long Josephson junction is investigated. The impurity has polarity and will be referred to as a dipole impurity because it also has a direction and, consequently, changes its sign under the space inversion transform x→−x. Such a model is used to describe the inductively coupled to the Josephson transmission line qubit and the misaligned Abrikosov vortex that penetrates into the long Josephson junction. We derive the approximate equations of motion for the fluxon centre of mass and its velocity. With the help of these equations we demonstrate that pinning and scattering of the fluxon on the impurity differs significantly from the case of the point impurity which is modelled by the derivative of the Dirac's δ-function.

The theoretical rationale for the use of contact heating when drying seeds in a vacuum

Purpose. The rationale for heating seeds by contact in a vacuum by analyzing the solutions of the differential equation of unsteady heat conduction. Methods. The differential heat equation, the Fourier method, the Sturm-Livull problem and the Dirac δ-function are used. Results. Based on the Fourier method and the Sturm-Livull problem, a differential heat equation is obtained, which makes it possible to determine the kinetics of seed heating depending on the thickness of the layer at a known temperature of the heating surface. A graphical analysis of the solution of the differential heat equation showed that for uniform drying of the seeds in vacuum it is necessary to apply mixing during drying. Conclusions. The analysis of the contact heating technological process, which was carried out on the basis of solving the differential equation of unsteady heat conduction, showed that the heating rate of seeds nonlinearly decreases with increasing layer thickness. This leads to uneven drying of the seeds in a vacuum. Therefore, the use of only one contact surface for heating seeds in a vacuum is impractical. Keywords: seeds, contact heating, heating temperature, layer thickness, vacuum, drying, heating.

Building confidence in the Dirac δ-function

In this note we present an example from undergraduate quantum mechanics designed to highlight the versatility of the Dirac δ-function. Namely, we compute the expectation value of the Hamiltonian of a free particle in a state described by a triangular wave function ψ(x). Since the first derivative of ψ(x) is piecewise constant, and because this Hamiltonian is proportional to the second order spatial derivative, students often end up finding the expectation value to be zero—an unphysical answer. This problem provides a pedagogical application of the Dirac δ-function. By arriving at the same result via alternate pathways, this exercise reinforces students’ confidence in the Dirac δ-function and highlights its efficiency and elegance.

Essential spectrum of Schrödinger operators with δ-interactions on unbounded hypersurfaces

Let Γ be a simply connected unbounded C 2-hypersurface in ℝ n such that Γ divides ℝ n into two unbounded domains D ±. We consider the essential spectrum of Schrodinger operators on ℝ n with surface δΓ-interactions which can be written formally as $${H_\Gamma } = - \Delta + W - {\alpha _\Gamma }{\delta _{\Gamma ,}}$$ , where −Δ is the nonnegative Laplacian in ℝ n , W ∈ L ∞(ℝ n ) is a real-valued electric potential, δΓ is the Dirac δ-function with the support on the hypersurface Γ and αΓ ∈ L ∞(Γ) is a real-valued coupling coefficient depending of the points of Γ. We realize H Γ as an unbounded operator A Γ in L 2(ℝ n ) generated by the Schrodinger operator $${H_\Gamma } = - \Delta + Won{\mathbb{R}^n}\backslash \Gamma $$ and Robin-type transmission conditions on the hypersurface Γ. We give a complete description of the essential spectrum of A Γ in terms of the limit operators generated by A Γ and the Robin transmission conditions.

Beta decay of nuclides 56Fe, 62Ni, 64Ni and 68Ni in the crust of magnetars

By introducing the Dirac δ-function and Pauli exclusion principle in the presence of superstrong magnetic fields (SMFs), we investigate the influence of SMFs on beta decay and the change rates of electron fraction (CREF) of nuclides 56Fe, 62Ni, 64Ni and 68Ni in magnetars, which are powered by magnetic field energy. We find that the magnetic fields have a great influence on the beta decay rates, and the beta decay rates can decrease by more than six orders of magnitude in the presence of SMFs. The CREF also decreases by more than seven orders of magnitude in the presence of SMFs.

On the deficiency index of the vector-valued Sturm–Liouville operator

Let R+:= [0, +∞), and let the matrix functions P, Q, and R of order n, n ∈ N, defined on the semiaxis R+ be such that P(x) is a nondegenerate matrix, P(x) and Q(x) are Hermitian matrices for x ∈ R+ and the elements of the matrix functions P−1, Q, and R are measurable on R+ and summable on each of its closed finite subintervals. We study the operators generated in the space Ln2(R+) by formal expressions of the form l[f] = −(P(f' − Rf))' − R*P(f' − Rf) + Qf and, as a particular case, operators generated by expressions of the form l[f] = −(P0f')' + i((Q0f)' + Q0f') + P'1f, where everywhere the derivatives are understood in the sense of distributions and P0, Q0, and P1 are Hermitianmatrix functions of order n with Lebesgue measurable elements such that P0−1 exists and ǁP0ǁ, ǁP0−1ǁ, ǁP0−1ǁǁP1ǁ2, ǁP0−1ǁǁQ0ǁ2 ∈ Lloc1(R+). Themain goal in this paper is to study of the deficiency index of the minimal operator L0 generated by expression l[f] in Ln2(R+) in terms of the matrix functions P, Q, and R (P0, Q0, and P1). The obtained results are applied to differential operators generated by expressions of the form \(l[f] = - f'' + \sum\limits_{k = 1}^{ + \infty } {{H_k}} \delta \left( {x - {x_k}} \right)f\), where xk, k = 1, 2,..., is an increasing sequence of positive numbers, with limk→+∞xk = +∞, Hk is a number Hermitian matrix of order n, and δ(x) is the Dirac δ-function.

A nonlocal diffusion model of a single species with age structure

Using theory of Fourier transformation, Dirac-δ function and von Foerster rule with nonlocal diffusion operator, we derive a single species population model with two age stages: juvenile and mature, which includes nonlocal dispersal both in diffusion and reaction. We found that the nonlocal reaction kernel function for mature population depends on the kernel function of diffusion for two age stages. This model takes the local diffusion model with only Laplacian operator or Laplacian operator combining with convection operator as particular examples. The existence, uniqueness, positivity and boundedness of initial value problem are also investigated in the present article. As a new model, its dynamics is worthy to be explored in the future work.

An new estimate of electron capture of nuclides 55Co and 56Ni in magnetars

Basing on the theory of relativity in ultrastrong magnetic fields (UMFs) and Random Phase Approximation (RPA), by using the Shell-Model Monte Carlo (SMMC) method, the influence of UMFS on electron capture (EC) and the rates of electron fraction change (REFC) of nuclides 55Co and 56Ni are investigated in magnetars. Basing on the Dirac δ-function and Pauli exclusion principle, we also discuss the influence on electron Fermi energy by UMFs in magnetars. The results show that the EC rates are increased greatly and even exceed five orders of magnitude by UMFs. The REFC is decreased even more than six orders of magnitude due to UMFs.

Magnetic islands and singular currents at rational surfaces in three-dimensional magnetohydrodynamic equilibria

Using the recently developed multiregion, relaxed MHD (MRxMHD) theory, which bridges the gap between Taylor's relaxation theory and ideal MHD, we provide a thorough analytical and numerical proof of the formation of singular currents at rational surfaces in non-axisymmetric ideal MHD equilibria. These include the force-free singular current density represented by a Dirac δ-function, which presumably prevents the formation of islands, and the Pfirsch-Schlüter 1/x singular current, which arises as a result of finite pressure gradient. An analytical model based on linearized MRxMHD is derived that can accurately (1) describe the formation of magnetic islands at resonant rational surfaces, (2) retrieve the ideal MHD limit where magnetic islands are shielded, and (3) compute the subsequent formation of singular currents. The analytical results are benchmarked against numerical simulations carried out with a fully nonlinear implementation of MRxMHD.

The effects of ultra-strong magnetic fields on electron capture rates for iron group nuclei in the outer crust of magnetars

Based on the work of Wang et al. (Chin. Phys. Lett. 29:049701, 2012), we re-investigated electron capture on iron group nuclei in the outer crust of magnetars and studied magnetar evolution. Effects of ultra-strong magnetic field on electron capture rates for 57Co have been analyzed in the nuclear shell model and under the Landau-level-quantization approximation, and the electron capture rates and the neutrino energy loss rates on iron group nuclei in the outer crust of magnetar have been calculated. The results show that electron capture rates on 57Co are increase greatly in the ultra-strong magnetic field, and above 3 orders of magnitude generally; and the neutrino energy loss rates by electron capture on iron group nuclei increase above 3 orders of magnitude in the range from B=4.414×1013 G to B=4.414×1015 G. These conclusions play an important role in future studying the evolution of magnetar. Furthermore, we modify the expressions of the electron chemical potential (Fermi energy) and phase space factor by introducing Dirac δ-function, and select appropriate parameters of temperature T, magnetic field B and matter density ρ in the our crust, thus our results will be reliable than those of Wang et al.