Gauss Law Research Articles

A discontinuity in the electromagnetic field of a uniformly accelerated charge

The electric field of a uniformly accelerated charge shows a plane of discontinuity, where the field extending only on one side of the plane, terminates abruptly on the plane with a finite value. This indicates a non-zero divergence of the electric field in a source-free region, implying a violation of Gauss law. In order to make the field compliant with Maxwell’s equations everywhere, an additional field component, proportional to a δ-function at the plane of discontinuity, is required. Such a ‘δ-field’ might be the electromagnetic field of the charge, moving with a uniform velocity approaching c, the speed of light, prior to the imposition of acceleration at infinity. However, a range of attempts to derive this δ-field for such a case, have not been entirely successful. Some of the claims of the derivation involve elaborate calculations with some not-so-obvious mathematical approximations. Since the result to be derived is already known from the constraint of its compliance with Maxwell’s equations, and the derivation involves the familiar text-book expressions for the field of a uniformly moving charge, one would expect an easy, simple approach, to lead to the correct result. Here, starting from the electromagnetic field of a uniformly accelerated charge in the instantaneous rest frame, in terms of the position and motion of the charge at the retarded time, we derive this δ-field, consistent with Maxwell’s equations, in a fairly simple manner. This is followed by a calculation of the energy in the δ-field, in an analytical manner without making any approximation, where we show that this energy is exactly the one that would be lost by the charge because of the radiation reaction on the charge, proportional to its rate of change of acceleration, that was imposed on it at a distant past.

Modifications of Schumann resonance spectra as an estimate of causative earthquake magnitude: The model treatment

Using a numerical model of the ELF radio wave scattering by seismogenic localized non-uniformity, we evaluate the earthquake (EQ) magnitude by using modifications in the power spectra of global electromagnetic (Schumann) resonance. The ionosphere non-uniformity is introduced as a disturbance in the vertical profile of atmosphere conductivity above the earthquake. Three types are considered of disturbance: the WHOLE, BOTTOM and TOP models, each depending on the EQ magnitude. The disturbance is axially symmetric; it varies along the radius according to the Gauss law with the scale corresponding to the EQ magnitude. The electromagnetic problem is solved by using the complex characteristic electric and magnetic heights relevant to conductivity profiles. These heights are found by solving the Riccati equation and are substituded into the two-dimensional telegraph equation (2DTE) for computing the vertical electric and two orthogonal magnetic fields in the Earth–ionosphere cavity. To avoid an impact of the field source coordinates, we apply the source model of independent lightning strokes uniformly distributed over the globe. The ionosphere non-uniformity is located above the North Pole, and the observer is positioned at the point of 63° N and 0° E, i.e., at 3 Mm distance from the disturbance. Comparison of power spectra in the regular and non-uniform cavities allowed us to single out their modifications at particular frequencies. Finally, equations are obtained connecting the EQ magnitude and the spectral modifications averaged over the frequency band covering four Schumann resonance modes. Computations indicate that reflections of radio waves from the localized seismogenic non-uniformity provide the noticeable impact when the EQ magnitude exceeds the M = 7 threshold. The BOTTOM model provides the highest impact on the Schumann resonance spectra, while the influence of the TOP model is the weakest one. The highest seismogenic effect exceeding the 20 dB level is observed in the meridional component HWE (the disturbance is found above the North Pole). Model data agree with available observations, and a further progress might be achieved in the case studies of particular EQs.

Piezoelectric galloping energy harvesting enhanced by topological equivalent aerodynamic design

The topological equivalent aerodynamic method is introduced to design a funnel-shape galloping energy harvester with wide working wind-speed range and high normalized harvesting power. The funnel shape bluff body is designed from the square bluff body to avoid the vortex reattachment, enhance the structural non-stream fluid flow and allow the pressure direction along the lift force. This design enlarges the aerodynamic force and thus improves the energy harvesting efficiency. To analyze the harvesting performance, the extended Hamilton principle and Gauss law are employed to derive the electro-mechanical coupled governing equations. The Galerkin procedure and equivalent structure method are then used to calculate the expressions of the onset galloping wind speed and harvested power density. Three energy harvesters of the square, triangular and funnel-shaped bluff bodies are tested in a closed direct-flow wind tunnel. The maximal experimental power densities for the funnel-shape, triangle and square bluff bodies are respectively 2.34 mW/cm3, 1.56 mW/cm3 and 0.207 mW/cm3. The corresponding experimental onset galloping wind speeds are 7 m/s, 9 m/s and 13 m/s. The good agreement between the theoretical prediction and experimental result demonstrates the accuracy of the mathematical model. The parameter analysis using the analytical model indicates that the energy harvester with funnel-shape bluff body always maintains the smallest onset speed and the highest power density. Besides, compared with previous studies of energy harvesting from the wind, the proposed energy harvester has the largest normalized power density. To better understand the fluid dynamics, two-dimensional unsteady numerical simulation is employed to show that the funnel shape has the largest lift coefficient compared to the other two cases. The velocity and pressure contours explain the physical cause that the flow pattern of the funnel shape can maintain the longest vortex region and highest vortex intensity.

Discrete aspects of continuous symmetries in the tensorial formulation of Abelian gauge theories

We show that standard identities and theorems for lattice models with $U(1)$ symmetry get re-expressed discretely in the tensorial formulation of these models. We explain the geometrical analogy between the continuous lattice equations of motion and the discrete selection rules of the tensors. We construct a gauge-invariant transfer matrix in arbitrary dimensions. We show the equivalence with its gauge-fixed version in a maximal temporal gauge and explain how a discrete Gauss's law is always enforced. We propose a noise-robust way to implement Gauss's law in arbitrary dimensions. We reformulate Noether's theorem for global, local, continuous or discrete Abelian symmetries: for each given symmetry, there is one corresponding tensor redundancy. We discuss semi-classical approximations for classical solutions with periodic boundary conditions in two solvable cases. We show the correspondence of their weak coupling limit with the tensor formulation after Poisson summation. We briefly discuss connections with other approaches and implications for quantum computing.

Solving Gauss's law on digital quantum computers with loop-string-hadron digitization

We show that using the loop-string-hadron (LSH) formulation of SU(2) lattice gauge theory (arXiv:1912.06133) as a basis for digital quantum computation easily solves an important problem of fundamental interest: implementing gauge invariance (or Gauss's law) exactly. We first discuss the structure of the LSH Hilbert space in $d$ spatial dimensions, its truncation, and its digitization with qubits. Error detection and mitigation in gauge theory simulations would benefit from physicality "oracles,'"so we decompose circuits that flag gauge invariant wavefunctions. We then analyze the logical qubit costs and entangling gate counts involved with the protocols. The LSH basis could save or cost more qubits than a Kogut-Susskind-type representation basis, depending on how the bases are digitized as well as the spatial dimension. The numerous other clear benefits encourage future studies into applying this framework.

Deviation of inverse square law based on Dunkl derivative: deformed Coulomb’s law

In this paper we consider the Coulomb’s law with deviation. We use the Dunkl derivative to derive the deformed Gauss law for the electric field and the electrostatic potentialwhich gives a new deformed electrostatics called a Dunkl-deformed electrostatics. Wemodify the Dunkl derivative for the electric field for multi sources or continuous chargedistribution. We discuss some examples of the Dunkl-deformed electrostatics.

An efficient, conservative, time-implicit solver for the fully kinetic arbitrary-species 1D-2V Vlasov-Ampère system

We consider the solution of the fully kinetic (including electrons) Vlasov-Ampère system in a one-dimensional physical space and two-dimensional velocity space (1D-2V) for an arbitrary number of species with a time-implicit Eulerian algorithm. The problem of velocity-space meshing for disparate thermal and bulk velocities is dealt with by an adaptive coordinate transformation of the Vlasov equation for each species, which is then discretized, including the resulting inertial terms. Mass, momentum, and energy are conserved, and Gauss's law is enforced to within the nonlinear convergence tolerance of the iterative solver through a set of nonlinear constraint functions while permitting significant flexibility in choosing discretizations in time, configuration, and velocity space. We mitigate the temporal stiffness introduced by, e.g., the plasma frequency through the use of high-order/low-order (HOLO) acceleration of the iterative implicit solver. We present several numerical results for canonical problems of varying degrees of complexity, including the multiscale ion-acoustic shock wave problem, which demonstrate the efficacy, accuracy, and efficiency of the scheme.

Loop, string, and hadron dynamics in SU(2) Hamiltonian lattice gauge theories

The question of how to efficiently formulate Hamiltonian gauge theories is experiencing renewed interest due to advances in building quantum simulation platforms. We introduce a reformulation of an SU(2) Hamiltonian lattice gauge theory---a loop-string-hadron (LSH) formulation---that describes dynamics directly in terms of its loop, string, and hadron degrees of freedom, while alleviating several disadvantages of quantumly simulating the Kogut-Susskind formulation. This LSH formulation transcends the local loop formulation of $d+1$-dimensional lattice gauge theories by incorporating staggered quarks, furnishing the algebra of gauge-singlet operators, and being used to reconstruct dynamics between states that have Gauss's law built in to them. LSH operators are then factored into products of "normalized" ladder operators and diagonal matrices, priming them for classical or quantum information processing. Self-contained expressions of the Hamiltonian are given up to $d=3$. The LSH formalism makes little use of structures specific to SU(2) and its conceptual clarity makes it an attractive approach to apply to other non-Abelian groups like SU(3).

Full-waveform inversion based on Kaniadakis statistics.

Full-waveform inversion (FWI) is a wave-equation-based methodology to estimate the subsurface physical parameters that honor the geologic structures. Classically, FWI is formulated as a local optimization problem, in which the misfit function, to be minimized, is based on the least-squares distance between the observed data and the modeled data (residuals or errors). From a probabilistic maximum-likelihood viewpoint, the minimization of the least-squares distance assumes a Gaussian distribution for the residuals, which obeys Gauss's error law. However, in real situations, the error is seldom Gaussian and therefore it is necessary to explore alternative misfit functions based on non-Gaussian error laws. In this way, starting from the κ-generalized exponential function, we propose a misfit function based on the κ-generalized Gaussian probability distribution, associated with the Kaniadakis statistics (or κ-statistics), which we call κ-FWI. In this study, we perform numerical simulations on a realistic acoustic velocity model, considering two noisy data scenarios. In the first one, we considered Gaussian noisy data, while in the second one, we considered realistic noisy data with outliers. The results show that the κ-FWI outperforms the least-squares FWI, providing better parameter estimation of the subsurface, especially in situations where the seismic data are very noisy and with outliers, independently of the κ-parameter. Although the κ-parameter does not affect the quality of the results, it is important for the fast convergence of FWI.

Emergence of Gauss' law in a Z2 lattice gauge theory in 1 + 1 dimensions

We explore a Z2 Hamiltonian lattice gauge theory in one spatial dimension with a coupling h, without imposing any Gauss' law constraint. We show that in our model h=0 is a free deconfined quantum critical point containing massless fermions where all Gauss' law sectors are equivalent. The coupling h is a relevant perturbation of this critical point and fermions become massive due to confinement and chiral symmetry breaking. To study the emergent Gauss' law sectors at low temperatures in this massive phase we use a quantum Monte Carlo method that samples configurations of the partition function written in a basis in which local conserved charges are diagonal. We find that two Gauss' law sectors, related by particle-hole symmetry, emerge naturally. When the system is doped with an extra particle, many more Gauss's law sectors related by translation invariance emerge. Using results in the range 0.01<h≤0.15 we find that three different mass scales of the model behave like hp where p≈0.579.

Central charges for the double coset

The state space of excited giant graviton brane systems is given by the Gauss graph operators. After restricting to the su(2|3) sector of the theory, we consider this state space. Our main result is the decomposition of this state space into irreducible representations of the su(2|2) ⋉ ℝ global symmetry. Excitations of the giant graviton branes are charged under a central extension of the global symmetry. The central extension generates gauge transformations so that the action of the central extension vanishes on physical states. Indeed, we explicitly demonstrate that the central charge is set to zero by the Gauss Law of the brane world volume gauge theory.

Ultra-wideband piezoelectric energy harvester based on Stockbridge damper and its application in smart grid

Transmission lines are often located in the field, and the long-term effective power supply for the online monitoring devices is an urgent problem for smart grid. To overcome this challenge, we propose a wide-band piezoelectric vibration energy harvester inspired by the Stockbridge damper in transmission lines system. The energy harvester has two subsystems, both of which are designed based on the 1:2 internal resonance principle. Thus, the harvester can exhibit up to four resonant frequencies. The geometric nonlinearity is introduced into the governing equations, and the coupled electrical circuit equation of this system is derived from the Gauss law. Theoretical simulations and experimental measurements show that the internal resonance can achieve a wider operating bandwidth. Based on the actual transmission line type and the measured wind speed data, the performance of energy harvester is analyzed. It demonstrates that the harvester can suppress the vortex-induced vibration of the transmission line within the wind velocity range of 1–4 m/s. The bandwidth of the internal resonance region can be up to 12.05 Hz. At the mean time, the harvest power for online monitoring devices could reach to 12.89 mW. The external noise is simulated by Gaussian white noise. The signal-to-noise ratio is introduced to consider the effect of the noise on the performance of the energy harvester. The results turn out that the harvester exhibits good robustness when the ratio is above 3 dB.

Ultra-broadband piezoelectric energy harvesting via bistable multi-hardening and multi-softening

An electromechanical coupled distributed parameter model is derived for a broadband piezoelectric energy harvester with nonlinear magnetic interaction and inductive–resistive interface circuit in the framework of the Hamilton’s principle and Gauss law. The approximate analytical solutions of the responses are obtained based on the equivalent mechanical representation and harmonic balance method. They are validated by experiment data and numerical simulations. The cubic-function discriminant of the analytical solution is introduced to determine the nonlinear boundaries of multiple solutions and the bandwidth with high harvested power. The stability of the multiple solutions is analyzed through Jacobi matrix of the modulation equation. The upward and downward sweep experiments exhibit the bistable and jump phenomena in the hardening range. The state plane of the modulation equation is used to show and explain why different initial conditions yield different stable dynamic motions and exhibit jump phenomenon. Multi-hardening and multi-softening nonlinearities are noted due to the multiply resonances by the inductance in the circuit and nonlinear characteristics of magnetic interaction in the structure. The analytical expression of the determinant of the nonlinear magnetic coefficient with double root of the response is derived to effectively characterize the observed phenomena. Different nonlinear types, e.g., typical nonlinear hardening and softening with two stable and one unstable solutions, and special nonlinear hardening and softening with one stable and one unstable solutions, are noted and investigated. The inductance and cubic magnetic coefficient affect the number and type of the nonlinearities. Multi-hardening or multi-softening nonlinearities enhance the performance of the piezoelectric energy harvester since its bandwidth is significantly broadened to cover up to 40 Hz in the low-frequency range.

Analysis of piezoelectric materials using sawtooth corrugation technique

Abstract This paper concentrates on piezoelectric material having circular attributes. Besides implication of the corrugation to this structure is brought out. Saw tooth type of corrugation applied to this piezoelectric material with respect to the host material and its inference is the highlight of this paper. Diaphragm uses in devices and its application in the field of materials is well established. How this type of arrangement holds physically under the circumstances of steady state form is the unique study under prevailing situation. Mechanical force acting on the perpendicular direction over the corrugated piezoelectric material paves the way for implementing the required physical terms. The relation among the stress, strain and the displacement of the diaphragm is brought out. How all of them behaves in an arrangement having a circular system of host material with that of the piezoelectric material is studied. Orthotropic plate model gets handy for this type of corrugation study. Application of potential to that of the piezoelectric material and the subsequent effect over that of the field intensity and flux density with reduced dielectric constant forms the focus of attention. With diaphragm being the focus of study analytical expression is arrived for the same. Application of Gauss law solves the problem further. Comprehensively treatment of the flexural rigidity by assigning fixed boundary conditions for this context forms the last part of this problem.

Improved Gauss law model and in-medium heavy quarkonium at finite density and velocity

We explore the in-medium properties of heavy-quarkonium states at finite baryo-chemical potential and finite transverse momentum based on a modern complex-valued potential model. Our starting point is a novel, rigorous derivation of the generalized Gauss law for in-medium quarkonium, combining the non-perturbative physics of the vacuum bound state with a weak coupling description of the medium degrees of freedom. Its relation to previous models in the literature is discussed. We show that our approach is able to reproduce the complex lattice QCD heavy quark potential even in the non-perturbative regime, using a single temperature dependent parameter, the Debye mass $m_D$. After vetting the Gauss-law potential with state-of-the-art lattice QCD data, we extend it to the regime of finite baryon density and finite velocity, currently inaccessible to first principles simulations. In-medium spectral functions computed from the Gauss-law potential are subsequently used to estimate the $\psi'/J/\psi$ ratio in heavy-ion collisions at different beam energies and transverse momenta. We find qualitative agreement with the predictions from the statistical model of hadronization for the $\sqrt{s_{NN}}$ dependence and a mild dependence on the transverse momentum.

Misuse of the minimal coupling to the electromagnetic field in quantum many-body systems

Consistency with the Maxwell equations determines how matter must be coupled to the electro-magnetic field (EMF) within the minimal coupling scheme. Specifically, if the Hamiltonian includes just a short-range repulsion among the conduction electrons, as is commonly the case for models of correlated metals, those electrons must be coupled to the full internal EMF, whose longitudinal and transverse components are self-consistently related to the electron charge and current densities through Gauss's and circuital laws, respectively. Since such self-consistency relation is hard to implement when modelling the non-equilibrium dynamics caused by the EMF, as in pump-probe experiments, it is common to replace in model calculations the internal EMF by the external one. Here we show that such replacement may be extremely dangerous, especially when the frequency of the external EMF is below the intra-band plasma edge.

Detecting Mind-Wandering from Eye Movement and Oculomotor Data during Learning Video Lecture

The purpose of this study was to detect mind-wandering experienced by pre-service teachers during a video learning lecture on physics. The lecture was videotaped and consisted of a live lecture in a classroom. The lecture was about Gauss's law on physics. We investigated whether oculomotor data and eye movements could be used as a marker to indicate the learner’s mind-wandering. Each data was collected in a study in which 24 pre-service teachers (16 females and 8 males) reported mind-wandering experience through self-caught method while learning physics video lecture during 30 minutes. A Tobii Pro Spectrum (sampling rate: 300 Hz) was used to capture their eye-gaze during learning Gauss's law through a course video. After watching the video lecture, we interviewed pre-service teachers about their mind-wandering experience. We first used the self-caught method to capture the mind-wandering timing of pre-service teachers while learning from video lectures. We detected more accurate mind-wandering segments by comparing fixation duration and saccade count. We investigated two types of oculomotor data (blink count, pupil size) and nine eye movements (average peak velocity of saccades; maximum peak velocity of saccades; standard deviation of peak velocity of saccades; average amplitude of saccades; maximum amplitude of saccades; total amplitude of saccades; saccade count/s; fixation duration; fixation dispersion). The result was that the blink count could not be used as a marker for mind-wandering during learning video lectures among them (oculomotor data and eye movements), unlike previous literatures. Based on the results of this study, we identified elements that can be used as mind-wandering markers while learning from video lectures that are similar to real classes, among the oculomotor data and eye movement mentioned in previous literatures. Additionally, we found that most participants focused on past thoughts and felt unpleasant after experiencing mind-wandering through interview analysis.

Fractons from Vector Gauge Theory.

Motivated by the prediction of fractonic topological defects in a quantum crystal, we utilize a reformulated elasticity duality to derive a description of a fracton phase in terms of coupled vector U(1) gauge theories. The fracton order and restricted mobility emerge as a result of an unusual Gauss law where electric field lines of one gauge field act as sources of charge for others. At low energies this vector gauge theory reduces to the previously studied fractonic symmetric tensor gauge theory. We construct the corresponding lattice model and a number of generalizations, which realize fracton phases via a condensation of stringlike excitations built out of charged particles, analogous to the p-string condensation mechanism of the gapped X-cube fracton phase.

Gauss law, minimal coupling and fermionic PEPS for lattice gauge theories

In these lecture notes, we review some recent works on Hamiltonian lattice gauge theories, that involve, in particular, tensor network methods. The results reviewed here are tailored together in a slightly different way from the one used in the contexts where they were first introduced. We look at the Gauss law from two different points of view: for the gauge field, it is a differential equation, while from the matter point of view, on the other hand, it is a simple, explicit algebraic equation. We will review and discuss what these two points of view allow and do not allow us to do, in terms of unitarily gauging a pure-matter theory and eliminating the matter from a gauge theory, and relate that to the construction of PEPS (Projected Entangled Pair States) for lattice gauge theories.

Polarization and relaxation effects induced by two-scale homogenization

Motivated by the two-scale convergence method introduced by G. Nguetseng and G. Allaire, we study the polarization and relaxation effects induced by two-scale homogenization. By virtue of Boltzmann–Poisson system, we characterize the microscopic properties of electrons in dielectric materials. The homogenized equations are obtained which describe the mean behaviors. By the homogenized equation, we get the polarization and relaxation effects which lead to the modified Gauss law; moreover, the dielectric function and the conductivity are also obtained.