Potential Equation Research Articles

Efficient Interpolation of Multilayer Periodic Green’s Functions with Electric and Magnetic Sources

A generalization of the efficient interpolation of periodic Green’s functions is presented for a multilayer medium hosting transverse electric current densities and transverse equivalent magnetic current densities at different interfaces. The mathematical model is realized in terms of Maxwell’s equations for multilayer media with isolated electric and magnetic equivalent current densities for large values of spectral variables or small values of spatial variables. This fact enables the use of Mixed Potential Integral Equation (MPIE) approaches in the spectral domain and provides asymptotic behaviors for Green’s functions of vector and scalar potentials for both electric and magnetic sources. Consequently, the singular behaviors of the Green’s functions around the source point are obtained as the spatial counterpart of the proposed spectral asymptotic behaviors. Thus, regularized multilayer periodic Green’s functions are obtained, which can be efficiently interpolated over the entire unit cell using Chebyshev’s polynomials.

Revisiting the Laplace Transform in Quantum Mechanics: Correcting a Flawed Approach for the Stationary Schr"odinger Equation

Abstract The Laplace transform is a valuable tool in physics, particularly in solving differential equations with initial or boundary conditions. A 2014 study by Tsaur and Wang (2014 \emph{Eur.~J.~Phys.} \textbf{35} 015006) introduced a Laplace-transform-based method to solve the stationary Schr"odinger equation for various potentials. However, their approach contains critical methodological flaws: the authors disregard essential boundary conditions and apply the residue theorem incorrectly in the inverse transformation process. These errors ultimately cancel out, leading to correct results despite a flawed derivation. In this paper, we revisit the use of the Laplace transform for the one-dimensional Schr"odinger equation, clarifying correct practices in handling boundary conditions and singularities. This analysis offers a sound and consistent framework for the application of Laplace transforms in stationary quantum mechanics, underscoring their educational utility in quantum mechanics coursework.

Zeta Potential Change and curing effects on Shield Muck solidified by Metal Ion Complex Catalysis

Using potentiometric testing, we investigated the zeta potential of shield muck curing materials’ particle surfaces, varying the concentration of metal ion complex. We analyzed the microscopic characteristics of shield muck curing products by using the electron microscopy, revealing the impact of metal ion complex on curing. Results showed that the metal ion complex significantly reduces the surface zeta potential of shield muck and conventional curing materials, with cement showing the most substantial effect, followed by shield muck, calcium carbonate, and calcium sulfate. The concentration of metal ion complex negatively correlates with the zeta potential within a range of 0 to 100 mmol/L. Comparing shield muck cured for 7 and 28 days, with and without metal ion complexes, indicated that metal ion complexes accelerate crystalline product formation, increasing the compressive strength of shield cinder cured products by over 35%. Fitted curves and empirical parameter equations of zeta potential and metal ion complex concentration align with experimental results. This study provides a reference for the efficient treatment and sustainable utilization of shield slag soil, holding significant practical engineering value.

Effect of pre-strain on the flux dynamics of current-carrying superconductors containing defects

This article presents a study that combines the electrostatic potential equation with the time-dependent Ginzburg–Landau equation, incorporating volume strain and utilizing Maxwell-type and current-carrying boundary conditions. Through the finite difference method, the research investigates the magnetic flux dynamics and energy variations in a type II superconductor with an artificial defect at the center, considering pre-strain, external magnetic field, and external current. The findings indicate that linear deformation and external electric field have a significant impact on the electrical properties, distribution of magnetic flux vortices, vortex dynamics, order parameter value, and overall total energy in superconductors with defects. These results contribute to a better understanding of the magnetic flux dynamics in superconductors under complex conditions.

Derivation and Physical Interpretation of the General Solutions to the Wave Equations for Electromagnetic Potentials

AbstractThe inhomogeneous wave equations for the scalar, vector, and Hertz potentials are derived starting from retarded charge, current, and polarization densities and then solved in the reciprocal (or k‐) space to obtain general solutions, which are formulated as nested integrals of such densities over the source volume, k‐space, and time. The solutions provide real‐space forms of the potentials and fields that are inherently free of singularities and do not require introduction by fiat of combinations of advanced and retarded terms as done previously to cure such singularities for the point‐charge model. Physical implications of these general solutions are discussed through specific examples involving either the real or reciprocal space forms of the different potentials. This approach allows for real‐ and reciprocal‐space expansions of potentials and fields for arbitrary distributions of charges and may lead to applications in condensed matter research and fluorescence‐based imaging.

Causal Structural Modeling of Survey Questionnaires via a Bootstrapped Ordinal Bayesian Network Approach

Abstract Survey questionnaires are commonly used by psychologists and social scientists to measure various latent traits of study subjects. Various causal inference methods such as the potential outcome framework and structural equation models have been used to infer causal effects. However, the majority of these methods assume the knowledge of true causal structure, which is unknown for many applications in psychological and social sciences. This calls for alternative causal approaches for analyzing such questionnaire data. Bayesian networks are a promising option as they do not require causal structure to be known a priori but learn it objectively from data. Although we have seen some recent successes of using Bayesian networks to discover causality for psychological questionnaire data, their techniques tend to suffer from causal non-identifiability with observational data. In this paper, we propose the use of a state-of-the-art Bayesian network that is proven to be fully identifiable for observational ordinal data. We develop a causal structure learning algorithm based on an asymptotically justified BIC score function, a hill-climbing search strategy, and the bootstrapping technique, which is able to not only identify a unique causal structure but also quantify the associated uncertainty. Using simulation studies, we demonstrate the power of the proposed learning algorithm by comparing it with alternative Bayesian network methods. For illustration, we consider a dataset from a psychological study of the functional relationships among the symptoms of obsessive-compulsive disorder and depression. Without any prior knowledge, the proposed algorithm reveals some plausible causal relationships. This paper is accompanied by a user-friendly open-source R package OrdCD on CRAN.

Global well-posedness of Boltzmann-Fermi-Dirac equation for hard potential

Global well-posedness of Boltzmann-Fermi-Dirac equation for hard potential

Main factors affecting polaraxis offset characteristics of intact spherical superconducting rotor

Superconducting rotor has great potential applications in the field of precision measurement due to its unique physical properties. The superconducting rotor magnetic levitation device can be used to fabricate high-precision angular velocity sensors. Under the action of external interference torque, the pole-axis deviation from the initial position is the cause of the superconducting rotor pole-axis drift error, in which the spherical surface error and the earth’s rotation belong to the main sources of error, and compensating for the pole-axis drift speed caused by the spherical surface error of the superconducting rotor is a key step in realizing the high-precision superconducting rotor magnetic levitation device. Based on this, the factors affecting the spherical surface error of a complete spherical superconducting rotor and the rotation of the earth on the pole-axis offset characteristics of a superconducting rotor are investigated. First, the magnetic support structure of the superconducting rotor is modeled based on the vector magnetic potential equation, the magnetic field strength distribution on the surface of the superconducting rotor in the ideal state (i.e. suspended in the center of the spherical cavity) is analyzed, and the magnetic support force characteristics are investigated. Then the magnetic support interference moment of the superconducting rotor caused by the spherical surface error is analyzed, and a superconducting rotor dynamics model is established based on the superconducting rotor dynamics equations, and the distribution law of the superconducting rotor pole-axis drift error under different rotor structural parameters is given. Finally, the influence of the earth’s rotation on the superconducting rotor drift test is investigated. The results provide a reference for subsequently improving rotor drift accuracy, optimizing rotor structure design and improving drift test methods.

Solutions of the Generalized Dunkl-Schrödinger Equation for Harmonic and Coulomb Potentials in two Dimensions

Solutions of the Generalized Dunkl-Schrödinger Equation for Harmonic and Coulomb Potentials in two Dimensions

Minimum energy principles in electrostatics

Abstract We employ the principle of minimum energy of the electrostatic field to derive Laplace’s equation for electric potential in charge-free space. We also apply the principle of minimum electrical potential energy of a system of discrete charges to explain the possibility of the existence of their stable equilibrium configuration on the surface of a conductor.

Multi-soliton, Breather, Lump and Their Interactions of the (2+1)-dimensional Generalized Potential Kadomtsev-Petviashvili Equation

In this paper, the generalized potential Kadomtsev-Petviashvili (gpKP) equation from fluid physics is systematic studied by some effective methods. Based on the Hirota bilinear method, multi-soliton solutions and multi-order breathers of the gpKP equation are presented. In the meanwhile, the multi-soliton can be transformed into the multi-order lumps under the long wave limit method. Furthermore, the interactions among the multi-soliton, multi-order breathers and multi-order lumps are discussed and simulated as well. The interaction between one-order lump and two solitons is elastic due to their shapes and amplitudes unchanged after collisions. The obtained results can immensely augment in understanding the nonlinear dynamic system deeply.

Bound state solutions of the Dunkl–Schrödinger equation for the sextic anharmonic oscillator potential

We consider the one-dimensional Schrödinger equation for the sextic anharmonic oscillator potential within the Dunkl formalism. Solutions of bound state type are constructed, and results are compared to the conventional scenario.

Exact solution of the Dunkl–Schrödinger equation for an angular-dependent potential and application of the Hellman–Feynman theorem

We determine the closed-form solution of a three-dimensional stationary Schrödinger equation for an angular-dependent potential in spherical coordinates within the Dunkl formalism. Furthermore, we calculate the change of the system’s total energy with respect to coupling constants in the potential by means of the Hellman–Feynman theorem.

Spectral and thermodynamic properties of a particle in multiparameter exponential-type radial potential

The solutions of the radial Schrödinger equation for a multiparameter potential of the exponential type are obtained using two traditional methods. The energy equation obtained was used to study both virial theorem and thermodynamic properties. The energy eigenvalues obtained for the potential and its subset potentials arecompletely bounded. The computed virial theorem depends on the parameters A, B, and V. The result also shows that the three parameters A, B, and V exhibit different features on the thermodynamic properties with V having the same variation for all the thermodynamic functions.

Conservative numerical algorithm for simulating thermoelectrical semiconductor device with unconditional optimal convergence analysis

We propose conservative type numerical method to simulate thermoelectrical semiconductor device problem, in which mixed finite element method used for electric potential equation, conservative characteristic finite element method for electron and hole concentration equations, and standard finite element method for heat conduction equation. By temporal-spatial error splitting argument, the optimal error estimates without certain time step restriction are derived, and low order convergence rate of electrostatic potential and electric field intensity will not affect the accuracy of the electron, hole density and temperature. Numerical tests are performed to validate the theoretical results and application performance of the given method.

A rotational ellipsoid model for solid Earth tide with high precision

Solid Earth tide represents the response of solid Earth to the lunar (solar) gravitational force. The yielding solid Earth due to the force has been thought to be a prolate ellipsoid since the time of Lord Kelvin, yet the ellipsoid’s geometry such as major semi-axis’s length, minor semi-axis’s length, and flattening remains unresolved. Additionally, the tidal displacement of reference point is conventionally resolved through a combination of expanded potential equations and given Earth model. Here we present a geometric model in which both the ellipsoid’s geometry and the tidal displacement of reference point can be resolved through a rotating ellipse with respect to the Moon (Sun). We test the geometric model using 23-year gravity data from 22 superconducting gravimeter (SG) stations and compare it with the current model recommended by the IERS (International Earth Rotation System) conventions (2010), the average Root Mean Square (RMS) deviation of the gravity change yielded by the geometric model against observation is 6.47 µGal (equivalent to 2.07 cm), while that yielded by the current model is 30.77 µGal (equivalent to 9.85 cm). The geometric model will greatly contribute to many application fields such as geodesy, geophysics, astronomy, and oceanography.

Determining plasma dose using equivalent total oxidation potential (ETOP): Concept to practical application via machine learning

Atmospheric pressure nonequilibrium plasma holds significant potential in biomedical applications due to its ability to generate reactive species at low temperatures. However, accurately quantifying and controlling plasma dosage remains challenging. Although equivalent total oxidation potential (ETOP) has been proposed for defining dosage, previous methods required measurement of various reactive oxygen and nitrogen species (RONS) densities, which are impractical in diverse plasma settings. Efficient ETOP prediction across variable conditions is thus essential. To address this, we propose a machine learning-based ETOP modeling method. This study collected RONS density data under various conditions using laser-induced fluorescence and trained an artificial neural network to predict ETOP values based on input parameters like voltage, gas flow rate, oxygen concentration, and humidity. This approach enables efficient ETOP prediction across variable conditions, supporting the standardization and clinical application of plasma medicine.

Iterative algorithm for computing magnetic field considering remanent magnetization and demagnetization

SUMMARY Efficient and high-precision methods for performing large-scale numerical calculations under high magnetic susceptibility conditions are essential for magnetic exploration applications. This paper proposes an algorithm for magnetic field modelling considering both demagnetization and remanent magnetization. The 3-D partial differential equation of magnetic potential is reduced to a 1-D ordinary differential equation in the space-wavenumber domain through a 2-D Fourier transform in the horizontal direction. Using the quadratic interpolation finite element method to obtain the pentagonal equation, the chasing method is used to solve the equation efficiently, and a contraction operator based on electromagnetic integral equation method is introduced to ensure stable iteration convergence. The validity of the algorithm was confirmed by comparing it with analytical solutions. Numerical examples demonstrate the importance of considering demagnetization under high magnetization conditions. Test results reveal that magnetic susceptibility is the sole factor affecting convergence among the factors examined. Further analysis showed that, for the same magnetic susceptibility, the algorithm converges with the same number of iterations and higher precision is achieved with an increasing number of nodes. Under the same grid configuration, higher magnetic susceptibility requires more iterations to converge and results in lower algorithm precision. The algorithm presented in this paper shows higher efficiency compared to those based on integral equations. Moreover, we demonstrate the algorithm's high accuracy when applied to models with anomalies near boundaries. Finally, the adaptability of the algorithm to complex geological models and undulating topography is verified by combining model tests with DEM data. This algorithm presents an efficient and high-precision method for magnetic field calculations under conditions of high susceptibility and strong remanent magnetization, offering promising prospects for geoscience applications.

Relativistic BEC extracted from a complex FRG flow equation

Abstract Based on the functional renormalization group (FRG) under the local potential approximation, we analyze the Bose–Einstein condensation (BEC) in the relativistic complex scalar theory. This framework leads to a complex flow equation of the effective potential, even with the well-known Litim regulator. In order to evaluate the condensate from such a complex effective potential, we impose a condition between chemical potential and mass, analogously to those in the free theory or the mean field theory. We elucidate that for the strongly (weakly) coupled theory, the phase diagrams computed from the FRG are more (less) deviated from that under the mean field approximation. This result implies that quantum fluctuations strongly affect the nonperturbative formation of the BEC.

Analytical prediction of electromagnetic performance for surface-embedded permanent magnet in-wheel machines considering iron’s nonlinearity

Accurate magnetic field calculation is the premise of electromagnetic performance prediction. Conventional subdomain (SD) techniques assume that the iron’s relative permeability is infinite, leading to falsely overestimated flux density. We propose an accurate magnetic field analytical model for permanent magnet (PM) in-wheel machines considering iron’s magnetization nonlinearity and saturation. Specifically, according to the excitation source and topology, the entire solution domain of the machine is divided into sub-regions such as stator slots/teeth, stator slot-openings/tooth-tips, air-gap, and rotor slots/teeth, etc. Poisson’s or Laplace’s magnetic vector potential (MVP) equations are solved using Maxwell’s electromagnetic theory and complex Fourier series methods in each sub-region. Specifically, in our approach, The Cauchy product theorem addresses the discontinuous magnetic permeability change at the slot and tooth interface. The machine’s magnetic saturation effect is considered by combining the actual magnetization characteristics of iron with an iterative algorithm. The general solution for the MVP is solved using the boundary conditions between adjacent subregions. Subsequently, electromagnetic properties such as air-gap flux density, back electromotive force (EMF), and electromagnetic torque are obtained. The accuracy of the analytical model is verified by finite element analysis (FEA) and prototype tests, which proved that the proposed analytical model can consider the iron’s nonlinearity and the magnetic saturation. In addition, the inaccurate overestimation of electromagnetic torque and air-gap magnetic flux density by the conventional SD techniques has also been proven.