Newtonian Potential Research Articles

Semi-analytical approximation of the normal derivative of the heat simple layer potential near the boundary of a two-dimensional domain

Semi-analytical approximation of the normal derivative of the heat simple layer potential near the boundary of a two-dimensional domain

Testing of Quadrature Formulas for the Direct Value of the Normal Derivative of a Single-Layer Potential at the Boundary of a Thin Body

Testing of Quadrature Formulas for the Direct Value of the Normal Derivative of a Single-Layer Potential at the Boundary of a Thin Body

Extended spherical harmonic expansion using multipolar bases of equivalent magnetic sources on arbitrary surfaces

PurposeThe purpose of this paper is to propose a compact model to represent the magnetic field outside the sources. This model provides the multipolar ordering of a spherical harmonic expansion far from the source while being valid in its close proximity.Design/methodology/approachThe authors investigate equivalent surface sources that enable to compute the field very close to any chosen surface that encloses the source. Then the authors present a method to find an appropriate initial basis and its associated inner product that allow to construct multipolar harmonic bases for these equivalent sources, where any vector of order k produces a field that decreases at least as fast as the field produced by a multipole of order k. Finally, those bases are numerically implemented to demonstrate their performances, both far from the source and in its close proximity.FindingsThe charge distribution and normal dipole distribution are well-suited to construct multipolar harmonic bases of equivalent sources. These bases can be described by as few parameters as the decreasing spherical harmonic expansion. Comparison with other numerical models shows its ability to compute the field both far from the source and close to it.Originality/valueA basis for normal dipole distribution has already been described in the literature. This paper presents a general method to construct a multipolar basis for equivalent sources and uses it to construct a basis for single-layer potential.

Jump relation for the conormal derivative of parabolic single layer potential

We consider a single layer potential generated by the fundamental solution of a second-order parabolic equation with Dini continuous coefficients. The potential is considered in a semibounded spatial domain with non-smooth lateral boundary from the Dini–Hölder class. We establish a jump relation for the conormal derivative of this potential at the lateral boundary of the domain.

Collapse of neutrino wave functions under Penrose gravitational reduction

Models of spontaneous wave function collapse have been postulated to address the measurement problem in quantum mechanics. Their primary function is to convert coherent quantum superpositions into incoherent ones, with the result that macroscopic objects cannot be placed into widely separated superpositions for observably prolonged times. Many of these processes will also lead to loss of coherence in neutrino oscillations, producing observable signatures in the flavor profile of neutrinos at long travel distances. The majority of studies of neutrino oscillation coherence to date have focused on variants of the continuous state localization model, whereby an effective decoherence strength parameter is used to model the rate of coherence loss with an assumed energy dependence. Another class of collapse models that have been proposed posit connections to the configuration of gravitational field accompanying the mass distribution associated with each wave function that is in the superposition. A particularly interesting and prescriptive model is Penrose’s description of gravitational collapse which proposes a decoherence time τ determined through Egτ∼ℏ, where Eg is a calculable function of the Newtonian gravitational potential. Here we explore application of the Penrose collapse model to neutrino oscillations, reinterpreting previous experimental limits on neutrino decoherence in terms of this model. We identify effects associated with both spatial collapse and momentum diffusion, finding that the latter is ruled out in data from the IceCube South Pole Neutrino Observatory so long as the neutrino wave packet width at production is σν,x≤2×10−12 m. Published by the American Physical Society 2024

Quasibound and quasinormal modes of a thick brane in Rastall gravity

In this work, we study the gravitational quasinormal modes of the thick brane in Rastall gravity. Using the asymptotic iteration and direct integration methods, we solve the quasinormal frequencies of the Rastall thick brane. We also obtained the waveforms of these quasinormal modes through numerical evolution. The results indicate that although the Rastall thick brane lacks a bound zero mode, when the Rastall parameter λ ≳ 0, a long-lived quasinormal mode appears. This long-lived quasinormal mode may restore the four-dimensional effective Newtonian potential on the brane on a large scale. This may provide a new perspective for the localization of gravity on thick branes, that a thick brane does not necessarily require the gravity to be localized, perhaps quasi-localized is sufficient.

An Induce-on-Boundary Magnetostatic Solver for Grid-Based Ferrofluids

This paper introduces a novel Induce-on-Boundary (IoB) solver designed to address the magnetostatic governing equations of ferrofluids. The IoB solver is based on a single-layer potential and utilizes only the surface point cloud of the object, offering a lightweight, fast, and accurate solution for calculating magnetic fields. Compared to existing methods, it eliminates the need for complex linear system solvers and maintains minimal computational complexities. Moreover, it can be seamlessly integrated into conventional fluid simulators without compromising boundary conditions. Through extensive theoretical analysis and experiments, we validate both the convergence and scalability of the IoB solver, achieving state-of-the-art performance. Additionally, a straightforward coupling approach is proposed and executed to showcase the solver's effectiveness when integrated into a grid-based fluid simulation pipeline, allowing for realistic simulations of representative ferrofluid instabilities.

A Causal Net approach to Relativistic Quantum Mechanics and Gravitation

In this paper we discuss a causal network model to describe quantum mechanics and gravittation, where space–time is built solely from point events and connecting probabilities. A causal net provides a representation which combines the quantum complementary features of both partticles and waves as each vertex on the causal net represents a possible point event or particle observation. Simultaneity on the causal net is defined by hyperplanes of equivalent proper time. The causal net model when constructed in Minkowksi space–time is shown to lead to a formulation for relativistic quantum mechanics including the Dirac equation. In a curved Riemann space–time the causal paths of the causal net are geodesics and in the local Lorentz frame the net probabilities and the Dirac formalism are preserved. The variation of space–time density of events in the causal paths modifies the metric and provides a space–time curvature leading to the Hilbert action associated with general relativity. Consideration of the Schwarzchild metric shows that in classical limit the perturbation in the density of possible events is proportional to the Newtonian gravitational potential.

On the Uniform Convergence of Approximations to the Tangential and Normal Derivatives of the Single-Layer Potential Near the Boundary of a Two-Dimensional Domain

On the Uniform Convergence of Approximations to the Tangential and Normal Derivatives of the Single-Layer Potential Near the Boundary of a Two-Dimensional Domain

The manipulation of photon blockade via Newtonian gravity

We theoretically investigate the model of a quadratically coupled optomechanical system with a Newtonian gravitational potential in the weak-driving regime, where the optical cavity is driven by an external laser. The steady state of the whole system is treated in the framework of a few-photon subspace. We find that the conventional single-photon blockade, nonstandard types of single-photon blockade, two-photon blockade, and photon-induced tunneling can be induced by gravity when the quadratic optomechanical coupling strength remains constant. Moreover, we find that gravitational potential energy can compensate for the lack of quadratic optomechanical coupling for observation photon blockade. In particular, the photon stream with super-Poissonian distribution can be converted into a sub-Poissonian, antibunching photon stream by changing the driving detuning when the gravitational potential energy is included. These results show that the gravity has potential for realizing the manipulation of photon blockade in a quadratically coupled optomechanical system.

Energies and a gravitational charge for massive particles in general relativity

In this paper, we investigate relations or differences among various conserved quantities which involve the matter Energy Momentum Tensor (EMT) in general relativity. These quantities include the energy with Einstein’s pseudo EMT, the generalized Komar integral, or the ADM energy, all of which can be derived from Noether’s second theorem, as well as an extra conserved charge recently proposed in general relativity. For detailed analyses, we apply definitions of these charges to a system of free massive particles. We employ the post-Newtonian (PN) expansion to make physical interpretations. We find that the generalized Komar integral is not conserved at the first non-trivial order in the PN expansion due to non-zero contributions at spatial boundaries, while the energy with Einstein’s pseudo EMT at this order agrees with a total energy of massive particles with gravitational interactions through the Newtonian potential, and thus is conserved. In addition, this total energy is shown to be identical to the ADM energy not only at this order but also all orders in the PN expansion. We next calculate an extra conserved charge for the system of massive particles, at all orders in the PN expansion, which turns out to be a total number of particles. We call it a gravitational charge, since it is clearly different from the total energy. We finally discuss an implication from a fact that there exist two conserved quantities, energy and gravitational charge, in general relativity.

Nonlocal Massive Gravity from Einstein Gravity.

We present a top-down construction of a three-dimensional nonlocal theory of massive gravity. This "nonlocal massive gravity" (NLMG) is obtained as the gravitational theory induced by Einstein gravity on a brane inserted in anti-de Sitter space modified by an overall minus sign. The theory involves an infinite series of increasingly complicated higher-derivative corrections to the Einstein-Hilbert action, with the quadratic term coinciding with new massive gravity. We obtain an analytic formula for the quadratic action of NLMG and show that its linearized spectrum consists of an infinite tower of positive-energy massive spin-2 modes. We compute the Newtonian potential and show that the introduction of the infinite series of terms makes it behave as ∼1/r at short distances, as opposed to the logarithmic behavior encountered when the series is truncated at any finite order. We use this and input from brane-world holography to argue that the theory may contain asymptotically flat black hole solutions.

Multiscale Modeling of Oxygen Evolution Through Generated Bilayer Porous Transport Layers for PEMWE Performance Improvement

Optimization of proton exchange membrane water electrolyzers (PEMWE) has become a focus of researchers looking for a reliable way to generate power. A vital component to PEMWE operation is the porous transport layer (PTL) on the anode side, which is where oxygen is produced. The PTL must allow water access to the catalyst layer and remove oxygen simultaneously. In this work, a previously developed imaging technique is used to generate bilayer PTL structures. A multiscale modeling approach was used to study the effect of a bilayer PTL on oxygen evolution and PEMWE performance. First, a micro scale model was used to predict oxygen transport pathways through different PTL structures. Results showed that the bilayer PTL results in higher oxygen saturation and faster oxygen transport through the PTL. Second, a macro scale model was used to predict performance using bilayer PTLs. Predictions showed potential values between 10 and 20 mV below single layer potential values. This points to the bilayer improving PEMWE operation. Findings from this work show how the addition of a mesoporous layer to a PTL substrate will improve oxygen transport and removal from the catalyst surface, which will improve PEMWE performance.

Testing Yukawa cosmology at the Milky Way and M31 galactic scales

We address the galaxy rotation curves through the Yukawa gravitational potential emerging as a correction of the Newtonian potential in extended theories of gravity. On the one hand, we consider the contribution of the galactic bulge, galactic disk, and the dark matter halo of the Navarro–Frenk–White profile, in the framework of the standard Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document}CDM model. On the other hand, we use modified Yukawa gravity to show that the rotational velocity of galaxies can be addressed successfully without the need for dark matter. In Yukawa gravity, we recover MOND and show that dark matter might be seen as an apparent effect due to the modification of the law of gravitation in terms of two parameters: the coupling constant α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document} and the characteristic length λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda $$\\end{document}. We thus test our theoretical scenario using the Milky Way and M31 rotation velocity curves. In particular, we place observational constraints on the free parameters of Yukawa cosmology through the Monte Carlo method and then compare our results with the predictions of the Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document}CDM paradigm by making use of Bayesian information criteria. Specifically, we find that λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda $$\\end{document} is constrained to be of the order of kpc, while cosmological data suggest λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda $$\\end{document} of the order of Gpc. To explain this discrepancy, we argue that there is a fundamental limitation in measuring λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda $$\\end{document} due to the role of quantum mechanics on cosmological scales.

Constraint free physics-informed machine learning for micromagnetic energy minimization

We introduce a novel method for micromagnetic energy minimization which uses physics-informed neural networks to find a magnetic configuration which minimizes the Gibbs Free energy functional without the need of any constraint optimization framework. The Cayley transform is applied to a neural network to assure that the model output lives on the Lie group of rotation matrices SO(3). For the stray field computation we use the splitting ansatz of Garcia-Cervera and Roma together with a hard constraint extreme learning machine in combination with a Taylor series approximation for very accurate evaluation of the single layer potential which only requires a very coarse discretization of the surface. Further, we present a modeling framework for constructive solid geometry which uses R-functions to exactly satisfy essential boundary conditions arising in the course of stray field computation. This framework can be applied to many other areas of interest. Our method shows promising results on the NIST μMAG Standard Problem #3, and is also applied to compute the demagnetization process of a hard magnetic Nd2Fe14B cube.

Boundary Effect Under 2D Newtonian Gravity Potential in the Phase Space

Boundary Effect Under 2D Newtonian Gravity Potential in the Phase Space

Plane strain problem of an elastic matrix containing multiple Gurtin–Murdoch material surfaces along straight segments

This paper presents the study of the plane strain problem of an infinite isotropic elastic medium subjected to far-field load and containing multiple Gurtin–Murdoch material surfaces located along straight segments. Each material segment represents a membrane of vanishing thickness characterized by its own elastic stiffness and residual surface tension. The governing equations, the jump conditions, and the surface tip conditions are reviewed. The displacements in the matrix are sought as the sum of complex variable single-layer elastic potentials whose densities are equal to the jumps in complex tractions across the segments. The densities are found by solving the system of coupled hypersingular boundary integral equations. The approximations by a series of Chebyshev’s polynomials of the second kind are used with the square root weight functions chosen to satisfy the tip conditions automatically. Numerical examples are presented to illustrate the influence of dimensionless parameters and to study the effects of interactions.

Thick Branes in Horndeski Gravity

We investigate thick brane solutions in the Horndeski gravity. In this setup, we found analytical solutions, applying the first-order formalism to two scalar fields where the first field comes from the nonminimal scalar-tensor coupling and the second is due to the matter contribution sector. With these analytical solutions, we evaluate the symmetric thick brane solutions in Horndeski gravity with four-dimensional geometry. In such a setup, we evaluate the gravity fluctuations to find “almost massless modes,” for any values of the Horndeski parameters. These modes were used to compute the corrections to the Newtonian potential and evaluate the limit four-dimensional gravity.

Improved Quadrature Formulas for the Direct Value of the Normal Derivative of a Single-Layer Potential

Improved Quadrature Formulas for the Direct Value of the Normal Derivative of a Single-Layer Potential

APPLICATION OF THE POTENTIAL THEORY TO SOLVING A MIXEDPROBLEM FOR A MULTIDIMENSIONAL ELLIPTIC EQUATION WITHONE SINGULAR COEFFICIENT

Potential theory has played a paramount role in both analysis and computation for boundary value problems for elliptic equations. In the middle of the last century, a potential theory was constructed for a two-dimensional elliptic equation with one singular coefficient. In the study of potentials, the properties of the fundamental solutions of the given equation are essentially and fruitfully used. At the present time, fundamental solutions of a multidimensional elliptic equation with one degeneration line are already known. In this paper, we investigate the double- and simple-layer potentials for this kind of elliptic equations. Results from potential theory allow us to represent the solution of the boundary value problems in integral equation form. By using some properties of Gaussian hypergeometric function, we prove limiting theorems and derive integral equations concerning a densities of the double- and simple-layer potentials. The obtained results are applied to find a solution of the mixed problem for the multidimensional singular elliptic equation in the half-space