Showing papers on "Dirac delta function published in 2018"
TL;DR: The post-Newtonian computations of the explicit analytic dynamics and motion of compact binaries are discussed within the most often applied Arnowitt–Deser–Misner formalism.
Abstract: Hamiltonian formalisms provide powerful tools for the computation of approximate analytic solutions of the Einstein field equations. The post-Newtonian computations of the explicit analytic dynamics and motion of compact binaries are discussed within the most often applied Arnowitt–Deser–Misner formalism. The obtention of autonomous Hamiltonians is achieved by the transition to Routhians. Order reduction of higher derivative Hamiltonians results in standard Hamiltonians. Tetrad representation of general relativity is introduced for the tackling of compact binaries with spinning components. Configurations are treated where the absolute values of the spin vectors can be considered constant. Compact objects are modeled by use of Dirac delta functions and their derivatives. Consistency is achieved through transition to d-dimensional space and application of dimensional regularization. At the fourth post-Newtonian level, tail contributions to the binding energy show up. The conservative spin-dependent dynamics finds explicit presentation in Hamiltonian form through next-to-next-to-leading-order spin–orbit and spin1–spin2 couplings and to leading-order in the cubic and quartic in spin interactions. The radiation reaction dynamics is presented explicitly through the third-and-half post-Newtonian order for spinless objects, and, for spinning bodies, to leading-order in the spin–orbit and spin1–spin2 couplings. The most important historical issues get pointed out.
102 citations
TL;DR: In this article, it was shown that a similar picture holds for generalizations of the SYK model with q-fermion interactions on a space whose metric is conformal to AdS2 × S1/Z2 and is subject to a non-trivial potential in addition to a delta function at the center of the interval.
Abstract: In [15] it was shown that the spectrum and bilocal propagator of SYK model with four fermion interactions can be realized as a three dimensional model in AdS2 ×S1/Z2 with nontrivial boundary conditions in the additional dimension. In this paper we show that a similar picture holds for generalizations of the SYK model with q-fermion interactions. The 3D realization is now given on a space whose metric is conformal to AdS2 × S1/Z2 and is subject to a non-trivial potential in addition to a delta function at the center of the interval. It is shown that a Horava-Witten compactification reproduces the exact SYK spectrum and a non-standard propagator between points which lie at the center of the interval exactly agrees with the bilocal propagator. As q → ∞, the wave function of one of the modes at the center of the interval vanish as 1/q, while the others vanish as 1/q2, in a way consistent with the fact that in the SYK model only one of the modes contributes to the bilocal propagator in this limit.
64 citations
TL;DR: A hyperbolic version of CQMOM is proposed that uses weighted Dirac delta functions and is able to control a symmetrical subset of the optimal moments in 2-D phase space.
42 citations
TL;DR: In this paper, the post-Newtonian computations of the explicit analytic dynamics and motion of compact binaries are discussed within the most often applied Arnowitt-Deser-Misner formalism.
Abstract: Hamiltonian formalisms provide powerful tools for the computation of approximate analytic solutions of the Einstein field equations. The post-Newtonian computations of the explicit analytic dynamics and motion of compact binaries are discussed within the most often applied Arnowitt-Deser-Misner formalism. The obtention of autonomous Hamiltonians is achieved by the transition to Routhians. Order reduction of higher derivative Hamiltonians results in standard Hamiltonians. Tetrad representation of general relativity is introduced for the tackling of compact binaries with spinning components. Configurations are treated where the absolute values of the spin vectors can be considered constant. Compact objects are modeled by use of Dirac delta functions and their derivatives. Consistency is achieved through transition to d-dimensional space and application of dimensional regularization. At the fourth post-Newtonian level, tail contributions to the binding energy show up. The conservative spin-dependent dynamics finds explicit presentation in Hamiltonian form through next-to-next-to-leading-order spin-orbit and spin1-spin2 couplings and to leading-order in the cubic and quartic in spin interactions. The radiation reaction dynamics is presented explicitly through the third-and-half post-Newtonian order for spinless objects, and, for spinning bodies, to leading-order in the spin-orbit and spin1-spin2 couplings. The most important historical issues get pointed out.
40 citations
TL;DR: This issue is tackled by formulating a well-posed and invertible integral transformation with Volterra and Fredholm terms that maps the Rijke system into a target system with desirable stability properties, allowing an explicit feedback control law to be derived.
35 citations
TL;DR: In this paper, the Riemann problem for the one-dimensional zero-pressure gas dynamics system is considered in the frame of α − solutions based on a solution concept defined in the setting of a product of distributions.
Abstract: The Riemann problem for the one-dimensional zero-pressure gas dynamics system is considered in the frame of α − solutions based on a solution concept defined in the setting of a product of distributions. The reformulated form of the zero-pressure gas dynamics system is provided and consequently the unique α − solution is obtained within a convenient class of distributions including the Dirac delta measure. It is shown that our constructed α − solution is reasonable compared with the known results using other methods. Furthermore, the result is generalized for the one-dimensional zero-pressure gas dynamics system with the Coulomb-like friction term, which enables us to see that the α − solution is not self-similar any more. It is shown that the time evolution of the delta shock wave discontinuity is represented by a parabolic curve under the influence of the Coulomb-like friction term.
34 citations
TL;DR: In this paper, the relativistic full Euler system with generalized Chaplygin proper energy density-pressure relation was studied, and the existence and uniqueness of delta shocks for the Riemann problem was established under the generalized Rankine-Hugoniot relation and entropy condition.
Abstract: The relativistic full Euler system with generalized Chaplygin proper energy density–pressure relation is studied. The Riemann problem is solved constructively. The delta shock wave arises in the Riemann solutions, provided that the initial data satisfy some certain conditions, although the system is strictly hyperbolic and the first and third characteristic fields are genuinely nonlinear, while the second one is linearly degenerate. There are five kinds of Riemann solutions, in which four only consist of a shock wave and a centered rarefaction wave or two shock waves or two centered rarefaction waves, and a contact discontinuity between the constant states (precisely speaking, the solutions consist in general of three waves), and the other involves delta shocks on which both the rest mass density and the proper energy density simultaneously contain the Dirac delta function. It is quite different from the previous ones on which only one state variable contains the Dirac delta function. The formation mechanism, generalized Rankine–Hugoniot relation and entropy condition are clarified for this type of delta shock wave. Under the generalized Rankine–Hugoniot relation and entropy condition, we establish the existence and uniqueness of solutions involving delta shocks for the Riemann problem.
23 citations
TL;DR: In this paper, the effective Boltzmann factor in a deformed formalism for modified Dirac delta, uniform, two-level and Gamma distributions is derived, and the effects of all parameters in the problems are calculated and shown graphically.
Abstract: In this article, after first introducing superstatistics, the effective Boltzmann factor in a deformed formalism for modified Dirac delta, uniform, two-level and Gamma distributions is derived. Then we make use of the superstatistics for four important problems in physics and the thermodynamic properties of the system are calculated. All results in the limit case are reduced to ordinary statistical mechanics. Furthermore, effects of all parameters in the problems are calculated and shown graphically.
19 citations
TL;DR: In this paper, a system of hyperbolic integro-differential equations for SH waves in a visco-elastic porous medium is considered and the inverse problem is to recover a kernel (memory) in the integral term of this system.
Abstract: We consider a system of hyperbolic integro-differential equations for SH waves in a visco-elastic porous medium The inverse problem is to recover a kernel (memory) in the integral term of this system We reduce this problem to solving a system of integral equations for the unknown functions We apply the principle of contraction mappings to this system in the space of continuous functions with a weight norm We prove the global unique solvability of the inverse problem and obtain a stability estimate of a solution of the inverse problem
19 citations
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TL;DR: In this paper, the authors focus on non-local conservation laws for vehicular traffic and show that the nonlocal-to-local limit can be rigorously justified provided the initial datum satisfies a one-sided Lipschitz condition and is bounded away from $0.
Abstract: Consider a nonlocal conservation where the flux function depends on the convolution of the solution with a given kernel. In the singular local limit obtained by letting the convolution kernel converge to the Dirac delta one formally recovers a conservation law. However, recent counter-examples show that in general the solutions of the nonlocal equations do not converge to a solution of the conservation law. In this work we focus on nonlocal conservation laws modeling vehicular traffic: in this case, the convolution kernel is anisotropic. We show that, under fairly general assumptions on the (anisotropic) convolution kernel, the nonlocal-to-local limit can be rigorously justified provided the initial datum satisfies a one-sided Lipschitz condition and is bounded away from $0$. We also exhibit a counter-example showing that, if the initial datum attains the value $0$, then there are severe obstructions to a convergence proof.
18 citations
TL;DR: The perturbation of an electric potential due to an insulating inclusion with corners is considered and the criterion for the existence of corner points on the inclusion boundary is determined in terms of the geometric factors.
Abstract: We consider the perturbation of an electric potential due to an insulating inclusion with corners. This perturbation is known to admit a multipole expansion whose coeffcients are linear combinations of generalized polarization tensors. We define new geometric factors of a simple planar domain in terms of a conformal mapping associated with the domain. The geometric factors share properties of the generalized polarization tensors and are the Fourier series coeffcients of a generalized external angle of the inclusion boundary. Since the generalized external angle contains the Dirac delta singularity at corner points, we can determine a criteria for the existence of corner points on the inclusion boundary in terms of the geometric factors. We illustrate and validate our results with numerical examples computed to a high degree of precision using integral equation techniques, the Nystrom discretization, and recursively compressed inverse preconditioning. (Less)
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TL;DR: In this article, the convergence of a nonlocal version of the Cahn-Hilliard equation to its local counterpart as the nonlocal convolution kernel is scaled using suitable approximations of a Dirac delta in a periodic boundary conditions setting is proved.
Abstract: In this paper we prove the convergence of a nonlocal version of the Cahn-Hilliard equation to its local counterpart as the nonlocal convolution kernel is scaled using suitable approximations of a Dirac delta in a periodic boundary conditions setting. This convergence result strongly relies on the dynamics of the problem. More precisely, the $H^{-1}$-gradient flow structure of the equation allows to deduce uniform $H^1$ estimates for solutions of the nonlocal Cahn-Hilliard equation and, together with a Poincare type inequality by Ponce, provides the compactness argument that allows to prove the convergence result.
TL;DR: In this paper, the effect of Fermi motion of constituents on the direct fragmentation of the $ J/\psi$ and $ \Upsilon$ states from the gluon using a light-cone wave function was studied.
Abstract: Fragmentation is the dominant mechanism for heavy quarkonia production with large transverse momentum. The study of heavy quarkonium production is a powerful tool to understand the dynamics of strong interactions. In this work, we study the effect of Fermi motion of constituents on the direct fragmentation of the $ J/\psi$
and $ \Upsilon$
states from the gluon using a light-cone wave function. Following this study, we compute the process-independent fragmentation functions (FFs) for a gluon to fragment into these bound states. Consistent with such a wave function we set up the kinematics of a gluon fragmenting into a quarkonium such that the Fermi motion of the constituents splits into longitudinal as well as transverse direction. In all previous calculations of heavy quarkonia FFs, by ignoring the Fermi motion of constituents, a delta function form was approximated for the meson wave function. Here, we present our numerical results for the $ g\rightarrow J/\psi$
and $ g\rightarrow \Upsilon$
FFs and show how the proposed meson wave function improves the previous results.
TL;DR: In this paper, the authors argue that such an approximation is improper in physics, because during the position measurement the energy transfer to the particle might be infinitely large, and that continuous and square-integrable functions of both sharp peak and sufficiently narrow width can then be better approximations of the Dirac delta function to represent the physical states of position.
Abstract: The Dirac delta function can be defined by the limitation of the rectangular function covering a unit area with decrease of the width of the rectangle to zero, and in quantum mechanics the eigenvectors of the position operator take the form of the delta function. When discussing the position measurement in quantum mechanics, one is prompted by the mathematical convention that uses the rectangular wave function of sufficiently narrow width to approximate the delta function in order to making the state of the position physical. We argue that such an approximation is improper in physics, because during the position measurement the energy transfer to the particle might be infinitely large. The continuous and square-integrable functions of both sharp peak and sufficiently narrow width can then be better approximations of the delta function to represent the physical states of position. When the slit experiment is taken as an apparatus of position measurement, no matter what potential is used to model the slit, only the ground state of the slit-dependent wave function matters.
TL;DR: In this paper, the authors have studied the scattering problem of relativistic fermions from a quaternionic double Dirac delta potential in the presence of the scalar and vector potentials in the quaternion formalism.
Abstract: We have studied the scattering problem of relativistic fermions from a quaternionic double Dirac delta potential. We have used Dirac equation in the presence of the scalar and vector potentials in the quaternionic formalism of relativistic quantum mechanics to study the problem. The wave functions of different regions have been derived. Then, using the reflection coefficient, transmission coefficient, and the continuity equation, the scattering problem has been investigated in detail. It has been shown that we have faced some fluctuations in the reflection and transmission coefficients.
TL;DR: In this paper, a combination of the implicit immersed boundary (IB) method and the lattice Boltzmann method (LBM) for a fluid-solid interaction involving an immersed body with moving boundaries and complex geometries is presented.
TL;DR: In this article, the Riemann solutions to a compressible fluid described by the generalized Chaplygin gas, where the external force is a constant, are given, and the location, velocity and weights of the delta shock wave are explicitly described.
Abstract: In this paper, we concern with the Riemann solutions to a compressible fluid described by the generalized Chaplygin gas, where the external force is a constant. Five exact solutions are given. In particular, the delta shock wave with a Dirac delta function in density and internal energy occurs in some solutions, and the location, velocity and weights of the delta shock wave are explicitly described. It is also noticed that because of the effect of the external force, these exact solutions are not self-similar.
TL;DR: In this article, the Klein-Gordon equation is extended in the presence of an Aharonov-Bohm magnetic field for the Cornell potential and corresponding wave functions as well as the spectra are obtained.
Abstract: The Klein–Gordon equation is extended in the presence of an Aharonov–Bohm magnetic field for the Cornell potential and the corresponding wave functions as well as the spectra are obtained. After in...
TL;DR: In this article, the spherically symmetric potential of the Dirac delta was generalized to the space of the free Hamiltonian and the spectrum of negative, zero and positive energy states was studied.
Abstract: The spherically symmetric potential $a \,\delta (r-r_0)+b\,\delta ' (r-r_0)$ is generalised for the $d$-dimensional space as a characterisation of a unique selfadjoint extension of the free Hamiltonian. For this extension of the Dirac delta, the spectrum of negative, zero and positive energy states is studied in $d\geq 2$, providing numerical results for the expectation value of the radius as a function of the free parameters of the potential. Remarkably, only if $d=2$ the $\delta$-$\delta'$ potential for arbitrary $a>0$ admits a bound state with zero angular momentum.
TL;DR: In this article, a Dirac delta term is introduced in the Schrodinger equation so that the quantum motion along the x-direction is constrained at y = 0, and the wave function is analyzed by using Green's function approach for several forms of the memory kernel, which are of particular interest.
Abstract: We investigate a generalized two-dimensional time-dependent Schrodinger equation on a comb with a memory kernel. A Dirac delta term is introduced in the Schrodinger equation so that the quantum motion along the x-direction is constrained at y = 0. The wave function is analyzed by using Green’s function approach for several forms of the memory kernel, which are of particular interest. Closed form solutions for the cases of Dirac delta and power-law memory kernels in terms of Fox H-function, as well as for a distributed order memory kernel, are obtained. Further, a nonlocal term is also introduced and investigated analytically. It is shown that the solution for such a case can be represented in terms of infinite series in Fox H-functions. Green’s functions for each of the considered cases are analyzed and plotted for the most representative ones. Anomalous diffusion signatures are evident from the presence of the power-law tails. The normalized Green’s functions obtained in this work are of broader interest...
TL;DR: In this paper, the weak coupling asymptotics were derived to order γ 2 of the ground state energy of the deltafunction Fermi gas and applied to a large class of finite convolution operators.
Abstract: Building on previous work of the authors, we here derive the weak coupling asymptotics to order γ2 of the ground state energy of the deltafunction Fermi gas. We use a method that can be applied to a large class of finite convolution operators.
TL;DR: In this article, the generalized Taylor's formula based on conformable derivatives and the neutrix limit was used to find the powers of the Dirac delta function for any δ-sequence.
Abstract: In this paper, starting from a fixed δ-sequence, we use the generalized Taylor’s formula based on conformable derivatives and the neutrix limit to find the powers of the Dirac delta function $\delta ^{r}$
and $(\delta^{\prime})^{r}$
for any $r\in \mathbb{R}$
.
TL;DR: Delta-M+ is an upgrade of the widely-used Delta-M method that truncates the forward scattering peak with a Dirac delta function, where the '+' symbol indicates that it essentially matches moments beyond the first M terms.
Abstract: The treatment of strongly anisotropic scattering phase functions is still a challenge for accurate radiance computations. The new Delta-M+ method resolves this problem by introducing a reliable, fast, accurate, and easy-to-use Legendre expansion of the scattering phase function with modified moments. Delta-M+ is an upgrade of the widely-used Delta-M method that truncates the forward scattering peak with a Dirac delta function, where the '+' symbol indicates that it essentially matches moments beyond the first M terms. Compared with the original Delta-M method, Delta-M+ has the same computational efficiency, but for radiance computations the accuracy and stability have been increased dramatically.
TL;DR: In this article, a volume average regularization for the second functional derivative operator appears in the metric-basis Wheeler-DeWitt equation is presented. But the regularization is not suitable for quantum general relativity.
Abstract: In this article, I present a volume average regularization for the second functional derivative operator that appears in the metric-basis Wheeler-DeWitt equation. Naively, the second functional derivative operator in the Wheeler-DeWitt equation is infinite, since it contains terms with a factor of a delta function or derivatives of the delta function. More precisely, the second functional derivative contains terms that are only well defined as a distribution---these terms only yield meaningful results when they appear within an integral. The second functional derivative may, therefore, be regularized by performing an integral average of the distributional terms over some finite volume; I argue that such a regularization is appropriate if one regards quantum general relativity (from which the Wheeler-DeWitt equation may be derived) to be the low-energy effective field theory of a full theory of quantum gravity. I also show that a volume average regularization can be viewed as a natural generalization of the same-variable second partial derivative for an ordinary multivariable function. Using the regularized second functional derivative operator, I construct an approximate solution to the Wheeler-DeWitt equation in the low-curvature, long-distance limit.
TL;DR: In this article, a generalized Rankine-Hugoniot relation and entropy condition for the delta shock wave with Dirac delta function in density were clarified. And the corresponding numerical results were also presented to illustrate their analysis.
Abstract: We concern with the Riemann problem the compressible fluid flow with the generalized Chaplygin gas. With the analysis on the phase plane, we rigorously confirm the occurrence of delta shock wave with Dirac delta function in density. Then the formation mechanism, generalized Rankine–Hugoniot relation and entropy condition for the delta shock wave are clarified. Based on these preparations, five kinds of exact solutions are obtained. Finally, the corresponding numerical results are also presented to illustrate our analysis.
TL;DR: Consejeria de Educacion, Junta de Castilla y Leon (VA057U16); Ministerio de Economia y Competitividad (MTM2014-57129-C2-1P); TUBITAK
Abstract: Consejeria de Educacion, Junta de Castilla y Leon (VA057U16); Ministerio de Economia y Competitividad (MTM2014-57129-C2-1-P); TUBITAK
TL;DR: In this paper, the Glauber-Sudarshan P-function was derived for a Schrodinger cat state in a surprisingly simple form, and it was shown that the properties of a generalized generalized delta function are different from those of a Dirac delta function and behave more like a pole in the complex plane.
Abstract: The Dirac delta function δ(x) is widely used in many areas of physics and mathematics. Here we consider the generalization of a Dirac delta function to allow the use of complex arguments. We show that the properties of a generalized delta function are very different from those of a Dirac delta function and that they behave more like a pole in the complex plane. We use the generalized delta function to derive the Glauber-Sudarshan P-function, P(α), for a Schrodinger cat state in a surprisingly simple form. Aside from their potential applications in classical electromagnetism and quantum optics, these results provide insight into the ability of the diagonal P-function to describe density operators with off-diagonal elements.
TL;DR: Higher order Legendre spectral element method is used for space discretization which is basically a domain decomposition method and it retains all the advantages of spectral and finite element methods.
TL;DR: In this article, the authors considered the propagation of initial disturbances in a one-dimensional medium that can vibrate and proposed a model that intermediates among the wave, heat, and transport equations.
Abstract: We present a model that intermediates among the wave, heat, and transport equations. The approach considers the propagation of initial disturbances in a one-dimensional medium that can vibrate. The medium is nonlinear in such a form that nonlocal differential expressions are required to describe the time evolution of solutions. Nonlocality was modeled with a space-time fractional differential equation of order 1 ≤ α ≤ 2 in time, and order 1 ≤ β ≤ 2 in space. We adopted the notion of Caputo for the time derivative and the Riesz pseudo-differential operator for the space derivative. The corresponding Cauchy problem was solved for zero initial velocity and initial disturbance, represented by either the Dirac delta or the Gaussian distributions. Well-known results for the conventional partial differential equations of wave propagation, diffusion, and (modified) transport processes were recovered as particular cases. In addition, regular solutions were found for the partial differential equation that arises from α = 2 and β = 1 . Unlike the above conventional cases, the latter equation permits the presence of nodes in its solutions.