Gauss Law Research Articles

Rank-2 toric code in two dimensions

We study a two-dimensional spin model obtained by "Higgsing" the rank-2 U(1) lattice gauge theory (LGT) with scalar or vector charges on the L_x * L_y square lattice under the periodic boundary condition (PBC). There are p degrees of freedom per orbital and three orbitals per unit cell in the spin model. The resulting spin model is a stabilizer code consisting of three mutually commuting projectors that are, in turn, obtained by Higgsing the mutually commuting Gauss's law operators and the magnetic field operators in the underlying LGT. The spin model thus obtained is exactly solvable, with the ground state degeneracy (GSD) D given by log_p D=2+(1+delta_{L_x mod p,0})(1+delta_{L_y mod p,0}) when p is a prime number. Two types of dipole excitations, pristine and emergent, are identified. Both the monopoles and the dipoles are free to move, with restrictions on monopoles to hop only by p lattice spacing along with certain directions. The monopole-monopole braiding phase depends on the separation of the x or y coordinates of the initial monopole positions, making it distinct from the ordinary anyon braiding statistics. The monopole-dipole braiding obeys the usual anyonic statistics. Despite the oddity, the monopole-monopole braiding phase can be understood as the Aharonov-Bohm phase of some emergent vector potentials.

Incident Plane-Wave Source Formulations for Leapfrog Complying-Divergence Implicit FDTD Method

The commonly used unconditionally stable finite-difference time-domain (FDTD) methods such as alternating direction implicit (ADI)-FDTD, and its one-step formulation, leapfrog ADI-FDTD, have been found to violate the divergence condition of Gauss's law. The recently proposed leapfrog complying-divergence implicit (CDI)-FDTD not only addresses this problem, but also features many advantages, including unconditional stability, minimal floating-point operations and one-step leapfrog update. To further expand its application, this paper presents the incident plane-wave source formulations for leapfrog CDI-FDTD. Two stable and efficient formulations with different advantages are presented for introducing the far-zone plane-wave source into the FDTD problem space, namely, the scattered-field (SF) formulation and total-field / scattered field (TF/SF) formulation. To deal with the discontinuity and inconsistency across TF/SF boundaries, the fields on the boundaries need special treatments with careful modifications to ensure stability and proper plane-wave injection. Numerical results show that the incident fields can be effectively injected into the problem space with the stability of leapfrog CDI-FDTD maintained in both formulations. In addition, comparisons of radar cross sections computed using leapfrog CDI-FDTD, leapfrog ADI-FDTD and explicit FDTD with both SF and TF/SF formulations are presented. These demonstrate the advantages of leapfrog CDI-FDTD method in solving far-zone plane-wave source problems, including high efficiency, unconditional stability and complying divergence.

Structure-Preserving and Helicity-Conserving Finite Element Approximations and Preconditioning for the Hall Mhd Equations

We develop structure-preserving finite element methods for the incompressible, resistive Hall magnetohydrodynamics (MHD) equations. These equations incorporate the Hall current term in Ohm's law and provide a more appropriate description of fully ionized plasmas than the standard MHD equations on length scales close to or smaller than the ion skin depth. We introduce a stationary discrete variational formulation of Hall MHD that enforces the magnetic Gauss's law exactly (up to solver tolerances) and prove the well-posedness and convergence of a Picard linearization. For the transient problem, we present time discretizations that preserve the energy and magnetic and hybrid helicity precisely in the ideal limit for two types of boundary conditions. Additionally, we present an augmented Lagrangian preconditioning technique for both the stationary and transient cases. We confirm our findings with several numerical experiments.

Inconsistency of islands in theories with long-range gravity

In ordinary gravitational theories, any local bulk operator in an entanglement wedge is accompanied by a long-range gravitational dressing that extends to the asymptotic part of the wedge. Islands are the only known examples of entanglement wedges that are disconnected from the asymptotic region of spacetime. In this paper, we show that the lack of an asymptotic region in islands creates a potential puzzle that involves the gravitational Gauss law, independently of whether or not there is a non-gravitational bath. In a theory with long-range gravity, the energy of an excitation localized to the island can be detected from outside the island, in contradiction with the principle that operators in an entanglement wedge should commute with operators from its complement. In several known examples, we show that this tension is resolved because islands appear in conjunction with a massive graviton. We also derive some additional consistency conditions that must be obeyed by islands in decoupled systems. Our arguments suggest that islands might not constitute consistent entanglement wedges in standard theories of massless gravity where the Gauss law applies.

SNNKIQ IMPROVING DIESEL ENGINE DIAGNOSTICS RELIABILITY AT UNSTEADY LOAD

The issues of rational use of energy in the operation of machine-tractor units by reducing downtime, including hidden ones, are very relevant. An increase in the energy efficiency of machine-tractor units can be achieved by diagnosing diesel engines according to functional parameters that are used as the power plant of unit. The difference in engine loading parameters during diagnosis and operation can be taken into account by correction factors for the tolerance of diagnostic parameters, which are determined for each brand of engine. (Research purpose) The research purpose is developing the methodology for determining correction coefficients for the tolerances of diagnostic parameters, taking into account the distribution of input effects. (Materials and methods) The values of the functional parameters of tractors in real operation significantly differ from the indicators in bench conditions. The probabilistic, functional transformation of random variables and computational methods were applied. The article proposes calculating the mathematical expectations of the functional parameters of the engines using the functional transformation of random variables and probabilistic characteristics of the input effect to simplify the process. (Results and discussion) Without taking into account the correction coefficients, the error in determining the deviations of functional parameters can be significant. Deviations of functional parameters from the values of the typical characteristic are maximal for the arcsin function and minimal for the Gauss law. The methodology for determining the correction coefficients for the tolerances of functional parameters was substantiated using the example of the D-240 engine. (Conclusions) The article presents quantitative characteristics of correction coefficients for the transition from operational tolerance to diagnostic. For most technological operations, diagnosing diesel engines with an unsteady load will reduce the error in monitoring the permissible values of functional parameters by 10-24 percent.

Gauge-invariant quantum circuits for U (1) and Yang-Mills lattice gauge theories

Quantum computation represents an emerging framework to solve lattice gauge theories (LGT) with arbitrary gauge groups, a general and long-standing problem in computational physics. While quantum computers may encode LGT using only polynomially increasing resources, a major openissue concerns the violation of gauge-invariance during the dynamics and the search for groundstates. Here, we propose a new class of parametrized quantum circuits that can represent states belonging only to the physical sector of the total Hilbert space. This class of circuits is compact yet flexible enough to be used as a variational ansatz to study ground state properties, as well as representing states originating from a real-time dynamics. Concerning the first application, the structure of the wavefunction ansatz guarantees the preservation of physical constraints such as the Gauss law along the entire optimization process, enabling reliable variational calculations. As for the second application, this class of quantum circuits can be used in combination with timedependent variational quantum algorithms, thus drastically reducing the resource requirements to access dynamical properties.

Investigation of piston ring wear engines by the mathematical statistics method

Improving the reliability and durability of internal combustion engines makes it possible to increase productivity and reduce the cost of operation. The study of nature and magnitude of piston rings wear is relevant and represents a scientific novelty. During the operation of internal combustion engines, it is almost impossible to determine the influence of various factors on the wear rate and the resource of piston rings. One of the most accurate ways to study the wear intensity and the service life values of internal combustion engines piston rings during operation is the statistical method. The method of mathematical statistics allows us to study the nature and magnitude of individual parts wear with a sufficient degree of accuracy, to determine the main causes of wear, to outline ways of increasing the parts wear resistance, to establish durability criteria, to reasonably assign standard service life of parts and plan the parts need for any period with a given probability, with climatic conditions being taking into account. The article presents the collected statistical material on the internal combustion engines piston rings wear of construction and road vehicles operated at construction sites of Rostov region. It has been processed according to the normal Gauss law and data on the wear and service life of piston rings are obtained. The obtained experimental data on the wear of piston rings in height are reliable to the distribution law and can be used to determine the intensity of engines piston rings wear at repair enterprises.

Quantum Electrostatics, Gauss’s Law, and a Product Picture for Quantum Electrodynamics; or, the Temporal Gauge Revised

We provide a theoretical foundation for the notion of the quantum coherent state of the electrostatic field of a static external charge distribution introduced in a 1998 paper and rederive formulae there for the inner products of a pair of such states. Contrary to what one might expect, these inner products are non-zero whenever the total charges of the two charge distributions are equal, even if the charge distributions themselves differ. We actually display two different frameworks for these same coherent states, in the second of which Gauss's law only holds in expectation value. We propose an experiment capable of ruling that out. The first framework leads to a 'product picture' for full QED -- i.e. a reformulation of standard QED in which it has a total Hamiltonian, arising as a sum of a free electromagnetic Hamiltonian, a free charged-matter Hamiltonian and an interaction term, acting on a 'physical subspace' of the full tensor product of charged-matter and electromagnetic-field Hilbert spaces. (The traditional Coulomb gauge formulation of QED isn't a product picture because, in it, the longitudinal part of the electric field is a function of the charged matter operators.) We do this for both Maxwell-Dirac and Maxwell-Schr\"odinger QED. For all states in the physical subspace of each of these systems, the charged matter is entangled with longitudinal photons and Gauss's law holds on the physical subspace as an operator equation; albeit the electric field operator and the Hamiltonian, while self-adjoint on the physical subspace, fail to be self-adjoint on the full tensor-product Hilbert space. Analogues of our coherent state inner products and of the product picture play a role in the author's matter-gravity entanglement hypothesis. Also, the product picture amounts to a temporal gauge quantization of QED which appears to be free from the difficulties of previous versions.

Perturbative F-theory 10-brane and M-theory 5-brane

The exceptional symmetry is realized perturbatively in F-theory which is the manifest U-duality theory. The SO(5) U-duality symmetry acts on both the 16 space-time coordinates and the 10 worldvolume coordinates. Closure of the Virasoro algebra requires the Gauss law constraints on the worldvolume. This set of current algebras describes a F-theory 10-brane. The SO(5) duality symmetry is enlarged to the SO(6) symmetry in the Lagrangian formulation. We propose actions of the F-theory 10-brane with SO(5) and SO(6) symmetries. The gauge fields of the latter action are coset elements of SO(6)/SO(6; ℂ) which include both the SO(5)/SO(5; ℂ) spacetime backgrounds and the worldvolume backgrounds. The SO(5) current algebra obtained from the Pasti-Sorokin-Tonin M5-brane Lagrangian leads to the theory behind M-theory, namely F-theory. We also propose an action of the perturbative M-theory 5-brane obtained by sectioning the worldvolume of the F-theory 10-brane.

A coupled formulation of fluid-structure interaction and piezoelectricity for modeling a multi-body energy harvester from vortex-induced vibrations

In this work, a coupled formulation for modeling multiple piezoelectric energy harvesters based on vortex-induced vibrations phenomenon at arbitrary locations was presented and experimentally validated. The mathematical formulation was performed by coupling the Navier-Stokes equations for incompressible fluid-flow, the Gauss law for piezoelectric equations and, a mass-spring-damper system for representing the oscillating rigid body. Piezoelectricity was described by a mixed formulation coupling mechanical and electrical variables. The coupled formulation was numerically implemented into a new OpenFOAM library. The library could be configured to add various piezoelectric materials for both types of wire connection, in series and in parallel. The numerical simulations for selected test cases were validated by experiments. After validating the numerical results, several working scenarios of three piezoelectric energy harvesters in tandem were simulated in order to evaluate the applicability of the proposed numerical formulation. The obtained results showed the synchronization state of the three harvesters operating in tandem was 500% wider than that of an isolated harvester. The interaction between cylinders played an important role in demonstrating that the output voltage of the three harvesters in tandem for a bimorph configuration connected in parallel (series) was approximately 128% (262%) higher than the unimorph one.

Continuous electric field modeling of Macro-Fiber Composites for actuation and energy harvesting

This paper presents a new low-order electric field model for Macro-Fiber Composite devices with interdigitated electrodes. Specifically, the paper proposes a continuous electric field model, where the differential form of Gauss's law is used to obtain spatial terms of the electric field, and the integral form of Gauss's law is used to obtain temporal term of the electric field. By neglecting three-dimensional effects, the method of matched asymptotic expansion is employed to obtain the electric field between two pairs of interdigitated electrodes under quasi-electrostatic conditions. By averaging the solution along with the thickness of piezoceramic fibers and expanding it to the device's entire geometry by Fourier series, the low-order model of the electric field between the interdigitated electrodes is obtained. The matched asymptotic expansion and the low-order model agree well with finite element results. Generalized Hamilton's principle is employed to obtain the coupled electromechanical equations of motion for a beam with the Macro-Fiber Composite device. The Euler-Bernoulli assumptions are used to approximate strain distribution. Under current assumptions, equivalent capacitance and electromechanical coupling terms are obtained without using discontinuous functions (e.g., Heaviside or piecewise functions) for electric field terms. The equivalent capacitance is within 2% of the values reported by the manufacturer. Frequency response functions are obtained for the output voltage, tip displacement, and optimum working load of the energy harvester, and the current and the tip displacement for the actuator by assumed modes solution for a cantilever beam. It is shown that the actuator's current is related to the optimum working load of the energy harvester by Ohm's law. In the limited case, the proposed continuous electric field model reduces to the constant electric field model. When the constant electric field model is compared to the continuous electric field model, it is shown that the constant electric field model overpredicts the equivalent capacitance and electromechanical coupling. The relative difference between the two models for equivalent capacitance is around 41%, and for electromechanical coupling is around 22%. These relative differences are in agreement with the empirical correction factors defined in the earlier constant field models.

Search for efficient formulations for Hamiltonian simulation of non-Abelian lattice gauge theories

Hamiltonian formulation of lattice gauge theories (LGTs) is the most natural framework for the purpose of quantum simulation, an area of research that is growing with advances in quantum-computing algorithms and hardware. It, therefore, remains an important task to identify the most accurate, while computationally economic, Hamiltonian formulation(s) in such theories, considering the necessary truncation imposed on the Hilbert space of gauge bosons with any finite computing resources. This paper is a first step toward addressing this question in the case of non-Abelian LGTs, which further require the imposition of non-Abelian Gauss's laws on the Hilbert space, introducing additional computational complexity. Focusing on the case of SU(2) LGT in 1+1 D coupled to matter, a number of different formulations of the original Kogut-Susskind framework are analyzed with regard to the dependence of the dimension of the physical Hilbert space on boundary conditions, system's size, and the cutoff on the excitations of gauge bosons. The impact of such dependencies on the accuracy of the spectrum and dynamics is examined, and the (classical) computational-resource requirements given these considerations are studied. Besides the well-known angular-momentum formulation of the theory, the cases of purely fermionic and purely bosonic formulations (with open boundary conditions), and the Loop-String-Hadron formulation are analyzed, along with a brief discussion of a Quantum Link Model of the same theory. Clear advantages are found in working with the Loop-String-Hadron framework which implements non-Abelian Gauss's laws a priori using a complete set of gauge-invariant operators. Although small lattices are studied in the numerical analysis of this work, and only the simplest algorithms are considered, a range of conclusions will be applicable to larger systems and potentially to higher dimensions.

Improving performance of piezoelectric energy harvesterunder electrostatic actuation using cavity

This study aims to investigate the effect of cavity on electric energy harvesting from cantilever beam vibrations under electrostatic actuation. Electrostatic actuation is created by a layer of radioisotope materials that is placed on the opposite side of the beam emitting electrons. When the beam is charged, the electrostatic force is generated between the beam and the opposite plate and pulls the beam towards itself. After the beam strikes the radioisotope, it is electrically discharged and then released. The piezoelectric layer converts the released microbeam vibration into electricity. The equations of motion coupled with the electrical effects of the piezoelectric layer are extracted using Hamilton's principle and Gauss's law. The equations are discretized by Galerkin method. The exact mode shape of the cantilever beam with the piezoelectric layer is employed as the comparison function. By identifying the relations governing the system, the output voltage and consequently the amount of harvested electrical energy are obtained using various parameters such as thickness and position of the cavity and system electrical resistance. The results indicates that creating cavity has a significant effect on the energy harvesting.

Searching for Magnetic Monopoles with Earth's Magnetic Field.

Magnetic monopoles have long been predicted in theory and could exist as a stable object in our Universe. As they move around in galaxies, magnetic monopoles could be captured by astrophysical objects like stars and planets. Here, we provide a novel method to search for magnetic monopoles by detecting the monopole moment of Earth's magnetic field. Using over six years of public geomagnetic field data obtained by the Swarm satellites, we apply Gauss's law to measure the total magnetic flux, which is proportional to the total magnetic charge inside Earth. To account for the secular variation of satellite altitudes, we define an altitude-rescaled magnetic flux to reduce the dominant magnetic dipole contribution. The measured magnetic flux is consistent with the existing magnetic field model that does not contain a monopole moment term. We therefore set an upper limit on the magnetic field strength at Earth's surface from magnetic monopoles to be |B_{m}|<0.13 nT at 95%confidence level, which is less than 2×10^{-6} of Earth's magnetic field strength. This constrains the abundance of magnetically charged objects, including magnetic black holes with large magnetic charges.

Electrohydrodynamics and thermal radiation effects on natural convection flow in an enclosed domain

The flow and heat transfer of electrohydrodynamics (EHD) natural convection in the annulus between a rectangular and a circular cylinder is studied numerically. The Rosseland diffusion approximation is used to account for the influence of heat radiation. The inner circular cylinder is considered to be heated and charged with a constant direct current (DC) voltage producing the electric field in the closed cavity. The fundamental Navier-Stokes equations and energy equation are coupled with Maxwell's and Gauss's law to establish the interaction between the velocity, temperature, and electric field for the dielectric fluid. While formulation, we presume that the charged fluid is ideal so that the effect of magnetic induction is negligible. A couple of transformations are used to convert the governing equations into the most appropriate form suitable for implementing the iterative finite difference scheme. According to the findings, the electric field and thermal radiations have a considerable impact on the flow field within the cavity. For instance, an increment in the charge diffusivity number, and the radiation parameter, the intensity of the streamlines, enhances significantly. The isotherms and isolines of electric charge density provide credible information about the heat flow and influence of charge injection. Moreover, the locally distributed Nusselt number at the inner and outer cylinders confirms that thermal radiation is the dominant mode of heat transfer.

Parametric study of a novel magneto-electro-aeroelastic energy harvesting system

This paper investigates a parametric study on a novel type of flutter-based aeroelastic harvester by utilizing magneto-electro-elastic materials, which can be used as an alternative to piezoelectric materials in aeroelastic harvesters to enhance the output electric power. The considered model includes a deformable clamped beam attached to a rigid airfoil, covering with one or two magneto-electro-elastic layers. Connected electrodes to magneto-electro-elastic layers capture potential electric from the produced electric field. Furthermore, an extrinsic coil is wrapped around the vibrating beam to produce the magnetic-based electric energy in the applied magneto-electro-elastic layer. First, the harvesting system is modeled as a discrete model to derive the governing equations using Faraday and Gauss laws, and the constitutive equations of magneto-electro-elastic materials according to Hamilton's principle. Then, different configurations have been examined and compared. Afterward, the effect of various parameters on the chosen harvester characteristics is investigated through considering stability to determine the optimal parameters resulting in the highest electric power generation. Lastly, the efficiency of the magneto-electro-aeroelastic harvester is compared with a piezo-aeroelastic counterpart indicating the supremacy of the presented harvester.

Gauss law in lattice QCD and its gauge-invariant Hilbert space

Gauss law in lattice QCD and its gauge-invariant Hilbert space

2-D–3-D Combined Numerical Method for Optimal Design of Magnetic Field Generator in Analytical Ferrograph

Analytical ferrograph (AF) is mainly used for the wear debris analysis in the lubrication oil of mechanical equipment, and the high-intensity magnetic field establishment of magnetic field generator (MFG) in AF is one of the key problems in design of AF. This article proposes an analytical, numerical, and experimental methodology for magnetic field design of AF. Based on Ohm's law and Gauss's flux law, the magnetic flux leakage coefficient of MFG is obtained, and then, the theoretical value of magnetic induction intensity at the working air gap is calculated, which provides a basis for the numerical method. Then, a 2-D finite-element model (FEM) is established to determine the magnetic circuit parameters of MFG, and a 3-D FEM is further developed to determine the ferrogram position parameters. Through the numerical calculations, the influence laws of magnetic circuit parameters and ferrogram position parameters on the magnetic field of MFG are obtained, and the optimal parameter ranges are determined. A Gauss meter is used for the magnetic field measurement, and the error between the experimental and numerical results is obtained to verify the correctness of the proposed numerical method. The average error between the measured and numerically calculated values is 6.07%. This article provides a positive contribution for wear particle analysis and the quick and accurate ferrogram preparation by AF.

Thermalization of Yang-Mills theory in a ( 3+1 )-dimensional small lattice system

We study the real-time evolution of SU(2) Yang-Mills theory in a ($3+1$)-dimensional small lattice system after interaction quench. We numerically solve the Schr\"odinger equation with the Kogut-Susskind Hamiltonian in the physical Hilbert space obtained by solving Gauss law constraints. We observe the thermalization of a Wilson loop to the canonical state; the relaxation time is insensitive to the coupling strength and estimated as ${\ensuremath{\tau}}_{\mathrm{eq}}\ensuremath{\sim}2\ensuremath{\pi}/T$ with temperatures $T$ at steady states. We also compute the vacuum persistence probability (the Loschmidt echo) to understand the relaxation from the dynamics of the wave function.

Noise and error analysis and optimization in particle-based kinetic plasma simulations

In this paper we analyze the noise in macro-particle methods used in plasma physics and fluid dynamics, leading to approaches for minimizing the total error, focusing on electrostatic models in one dimension. We begin by describing kernel density estimation for continuous values of the spatial variable x, expressing the kernel in a form in which its shape and width are represented separately. The covariance matrix C(x,y) of the noise in the density is computed, first for uniform true density. The bandwidth of the covariance matrix is related to the width of the kernel. A feature that stands out is the presence of constant negative terms in the elements of the covariance matrix both on and off-diagonal. These negative correlations are related to the fact that the total number of particles is fixed at each time step; they also lead to the property ∫C(x,y)dy=0. We investigate the effect of these negative correlations on the electric field computed by Gauss's law, finding that the noise in the electric field is related to a process called the Ornstein-Uhlenbeck bridge, leading to a covariance matrix of the electric field with variance significantly reduced relative to that of a Brownian process.For non-constant density, ρ(x), still with continuous x, we analyze the total error in the density estimation and discuss it in terms of bias-variance optimization (BVO). For some characteristic length l, determined by the density and its second derivative, and kernel width h, having too few particles within h leads to too much variance; for h that is large relative to l, there is too much smoothing of the density. The optimum between these two limits is found by BVO. For kernels of the same width, it is shown that this optimum (minimum) is weakly sensitive to the kernel shape.We repeat the analysis for x discretized on a grid. In this case the charge deposition rule is determined by a particle shape. An important property to be respected in the discrete system is the exact preservation of total charge on the grid; this property is necessary to ensure that the electric field is equal at both ends, consistent with periodic boundary conditions. We find that if the particle shapes satisfy a partition of unity property, the particle charge deposited on the grid is conserved exactly. Further, if the particle shape is expressed as the convolution of a kernel with another kernel that satisfies the partition of unity, then the particle shape obeys the partition of unity. This property holds for kernels of arbitrary width, including widths that are not integer multiples of the grid spacing.We show results relaxing the approximations used to do BVO optimization analytically, by doing numerical computations of the total error as a function of the kernel width, on a grid in x. The comparison between numerical and analytical results shows good agreement over a range of particle shapes.We discuss the practical implications of our results, including the criteria for design and implementation of computationally efficient particle shapes that take advantage of the developed theory.