Gauss Law Research Articles

Constructing Berry-Maxwell equations with Lorentz invariance and Gauss's law of Weyl monopoles in four-dimensional energy-momentum space

Constructing Berry-Maxwell equations with Lorentz invariance and Gauss's law of Weyl monopoles in four-dimensional energy-momentum space

Application Of Gauss's Law and Biot-Savart Law to Charged Bodies with Inhomogeneous Charges

Application Of Gauss's Law and Biot-Savart Law to Charged Bodies with Inhomogeneous Charges

Six-wave interaction in multimode waveguides with Kerr nonlinearity with allowance for the Gaussian structure of pump waves

The quality of wavefront conjugation in the case of six-wave interaction in two-dimensional multimode waveguides with Kerr nonlinearity is analyzed under the condition that one of the pump waves excites a zero waveguide mode and the amplitude distribution of the other pump wave at the waveguide end facet varies according to the Gauss law. It is shown that in a waveguide with infinitely conducting walls, the half-width of the modulus of the point spread function of a six-wave radiation converter is completely determined by the transverse size of the waveguide and weakly depends on the width of the Gaussian pump wave. In a parabolic-index profile waveguide, a decrease in the width of the Gaussian pump wave at the waveguide ends commonly leads to a monotonic decrease in the half-width of the point spread function modulus.

Constraints on the topology of Type IIB string theory

We discuss some topological constraints on Type IIB string theory that cannot be described by elementary characteristic classes. Anomaly cancellation of the worldvolume theory of a D3-brane requires a shifted Dirac quantization condition of the Ramond-Ramond 5-form flux. However, the required shift is sometimes inconsistent depending on the topology of spacetime backgrounds. The obstruction to the existence of a shifted quantization is described by a degree-6 cohomology class whose definition involves spin structure of spacetime manifolds. The cohomology class is interpreted as a background D3-charge, and the Gauss law constraint requires inclusion of additional D3-branes to cancel it.

Free vibration of functionally-graded piezoelectric semiconductor rectangular beam under thermal load

In this paper, the free vibration of a functionally-graded piezoelectric semiconductor (FGPS) rectangular beam under thermal load is studied within the framework of the theory of Timoshenko beam and the theory of piezoelectric semiconductor. It is assumed that the material properties in the functional gradient layer change smoothly. The multi-field coupling in the beam is deduced by introducing the first-order shear deformation theory in this paper, and the governing equations including motion equation, Gauss law and current continuity condition are obtained. The vibration of a simply supported FGPS rectangular beam is realized by solving the governing equations. In numerical examples, the natural frequencies of the FGPS rectangular beam under thermal load are analyzed through different perspectives. The results show that for different length scales, the natural frequency of the FGPS beam under thermal load is reduced by increasing the initial electron concentration in a certain range. The damping characteristics are insensitive to temperature change, and the damping characteristics are decreased by increasing length to thickness ratio and the thickness of the FGPS layer.

Classification of classical spin liquids: Typology and resulting landscape

Classical spin liquids (CSL) lack long-range magnetic order and are characterized by an extensive ground-state degeneracy. We propose a classification scheme of CSLs based on the structure of the flat bands of their Hamiltonians. Depending on absence or presence of the gap from the flat band, the CSL are classified as algebraic or fragile topological, respectively. Each category is further classified: the algebraic case by the nature of the emergent Gauss's law at the gap-closing point(s), and the fragile topological case by the homotopy of the eigenvector winding around the Brillouin zone. Previously identified models of CSLs fit snugly into our scheme, on a landscape where algebraic CSLs are located at transitions between fragile topological ones. It also allows us to present new families of models illustrating this landscape, which hosts both fragile topological and algebraic CSLs, as well as transitions between them. Published by the American Physical Society 2024

Entanglement entropy and the boundary action of edge modes

We consider an antisymmetric gauge field in the Minkowski space of d-dimension and decompose it in terms of the antisymmetric tensor harmonics and fix the gauge. The Gauss law implies that the normal component of the field strength on the spherical entangling surface will label the superselection sectors. From the two-point function of the field strength on the sphere, we evaluate the logarithmic divergent term of the entanglement entropy of edge modes of p-form field. We observe that the logarithmic divergent term in entanglement entropy of edge modes coincides with the edge partition function of co-exact p-form on the sphere when expressed in terms of the Harish-Chandra characters. We also develop a boundary path integral of the antisymmetric p-form gauge field. From the boundary path integral, we show that the edge mode partition function corresponds to the co-exact (p − 1)-forms on the boundary. This boundary path integral agrees with the direct evaluation of the entanglement entropy of edge modes extracted from the two-point function of the normal component of the field strength on the entangling surface.

Wegner's Ising gauge spins versus Kitaev's Majorana partons: Mapping and application to anisotropic confinement in spin-orbital liquids

Emergent gauge theories take a prominent role in the description of quantum matter, supporting deconfined phases with topological order and fractionalized excitations. A common construction of \mathbb{Z}_2ℤ2 lattice gauge theories, first introduced by Wegner, involves Ising gauge spins placed on links and subject to a discrete \mathbb{Z}_2ℤ2 Gauss law constraint. As shown by Kitaev, \mathbb{Z}_2ℤ2 lattice gauge theories also emerge in the exact solution of certain spin systems with bond-dependent interactions. In this context, the \mathbb{Z}_2ℤ2 gauge field is constructed from Majorana fermions, with gauge constraints given by the parity of Majorana fermions on each site. In this work, we provide an explicit Jordan-Wigner transformation that maps between these two formulations on the square lattice, where the Kitaev-type gauge theory emerges as the exact solution of a spin-orbital (Kugel-Khomskii) Hamiltonian. We then apply our mapping to study local perturbations to the spin-orbital Hamiltonian, which correspond to anisotropic interactions between electric-field variables in the \mathbb{Z}_2ℤ2 gauge theory. These are shown to induce anisotropic confinement that is characterized by emergence of weakly-coupled one-dimensional spin chains. We study the nature of these phases and corresponding confinement transitions in both absence and presence of itinerant fermionic matter degrees of freedom. Finally, we discuss how our mapping can be applied to the Kitaev spin-1/2 model on the honeycomb lattice.

Development of a MATLAB‐based graphical user interface to illustrate Coulomb's law and Gauss's law

AbstractCoulomb's law and Gauss's law are two fundamental laws in electrostatics that involve electric charges and electric fields. These concepts that are not visible physically in real life can be challenging for students to understand, especially when it involves complex charge distributions. This paper presents a MATLAB‐based graphical user interface (GUI) to visualise the concept of Coulomb's law and Gauss's law as well as a serious snake game to reinforce users' knowledge. This MATLAB GUI features a diverse range of 3‐dimensional (3D) visualisations illustrating electric field attributes such as electric force, charge distribution and electric flux. These visualisations dynamically adapt to user‐defined parameters, offering a rich and varied exploration of the electric field's characteristics without confining the scope to a specific count of models. In Coulomb's law section, the GUI plots the electric forces in 3‐D for dipole, three‐point charges, and unlimited charge. The Gauss's law section consists of windows for illustration of the fundamentals of Gauss's law for the electric field, and the illustration for the application of Gauss's law in finding the electric field for a point charge, infinite line charge and infinite plane charge. The serious snake game developed in this work allows students to engage in quizzes related to Coulomb's law and Gauss's law to serve as a tool for reinforcing their learning outcomes. The MATLAB‐based GUI was chosen as the platform due to its excellent visualisation capabilities, ease of programming and capability to act as a powerful tool to enhance the learning of Coulomb's law and Gauss's law for students.

Holography and localization of information in quantum gravity

Within the AdS/CFT correspondence, we identify a class of CFT operators which represent diff-invariant and approximately local observables in the gravitational dual. Provided that the bulk state breaks all asymptotic symmetries, we show that these operators commute to all orders in 1/N with asymptotic charges, thus resolving an apparent tension between locality in perturbative quantum gravity and the gravitational Gauss law. The interpretation of these observables is that they are not gravitationally dressed with respect to the boundary, but instead to features of the state. We also provide evidence that there are bulk observables whose commutator vanishes to all orders in 1/N with the entire algebra of single-trace operators defined in a space-like separated time-band. This implies that in a large N holographic CFT, the algebra generated by single-trace operators in a short-enough time-band has a non-trivial commutant when acting on states which break the symmetries. It also implies that information deep in the interior of the bulk is invisible to single-trace correlators in the time-band and hence that it is possible to localize information in perturbative quantum gravity.

Classification of classical spin liquids: Detailed formalism and suite of examples

The hallmark of highly frustrated systems is the presence of many states close in energy to the ground state. Fluctuations between these states can preclude the emergence of any form of order and lead to the appearance of spin liquids. Even on the classical level, spin liquids are not all alike: they may have algebraic or exponential correlation decay, and various forms of long wavelength description, including vector or tensor gauge theories. Here, we introduce a classification scheme, allowing us to fit the diversity of classical spin liquids (CSLs) into a general framework as well as predict and construct new kinds. CSLs with either algebraic or exponential correlation-decay can be classified via the properties of the bottom flat band(s) in their soft-spin Hamiltonians. The classification of the former is based on the algebraic structures of gapless points in the spectra, which relate directly to the emergent generalized Gauss's laws that control the low-temperature physics. The second category of CSLs, meanwhile, are classified by the fragile topology of the gapped bottom band(s). Utilizing the classification scheme we construct new models realizing exotic CSLs, including one with anisotropic generalized Gauss's laws and charges with subdimensional mobility, one with a network of pinch-line singularities in its correlation functions and a series of fragile topological CSLs connected by zero-temperature transitions. Published by the American Physical Society 2024

Lifting Topological Codes: Three-Dimensional Subsystem Codes from Two-Dimensional Anyon Models

Topological subsystem codes in three spatial dimensions allow for quantum error correction with no time overhead, even in the presence of measurement noise. The physical origins of this single-shot property remain elusive, in part due to the scarcity of known models. To address this challenge, we provide a systematic construction of a class of topological subsystem codes in three dimensions built from Abelian quantum double models in one fewer dimension. Our construction not only generalizes the recently introduced subsystem toric code [Kubica and Vasmer, Nat. Commun. , 6272 (2022)] but also provides a new perspective on several aspects of the original model, including the origin of the Gauss law for gauge flux, and boundary conditions for the code family. We then numerically study the performance of the first few codes in this class against phenomenological noise to verify their single-shot property. Lastly, we discuss Hamiltonians naturally associated with these codes, and argue that they may be gapless. Published by the American Physical Society 2024

Features of calculating the strength of gantry and ship cranes gear units in the probabilistic aspect

The performed study is the result of collecting statistical data on failures of port transshipment and floating cranes. It is found that many accidents of components and parts of crane mechanisms can be associated with an erroneous determination of design loads. In order to increase the strength reliability, numerous field tests were carried out in river and sea ports under various modes of transshipment for different types of cargo. Tests have shown that operational loads are random in nature, and their laws of distribution over time of individual crane mechanisms can be approximated by truncated normal Gauss laws, accounting for which at the stage of new design developments, as shown by the practice of performing strength fatigue calculations, is not possible. In the article, in order to use the obtained laws at the design stage, it is proposed to replace the representation of operational loads with initial moments that are common to the entire crane mechanism, and transfer individual characteristics consideration such as stress concentration coefficients, fatigue curve parameters, operating time or test base, to subsequent stages of machine components development. For practical testing of the operational loads representation by the normal Gauss laws, a critical analysis of their initial moments, which makes it possible to identify inconsistencies in the representations of variable loads proposed for strength calculation, is given. For the corrected truncated normal load distribution law, the initial moments, the value of the order of which corresponds to the formulas structure used in calculating fatigue resistance at variable load amplitudes (in particular, under alternating loading modes), are determined. The presentation of corrected normal distribution laws with initial moments complements the normative documents used in practice when performing strength calculations under variable loading conditions, significantly simplifies calculations and makes it possible to take it into account at the stage of new design developments in accordance with the formulas established in the current normative documents.

General covariance and dynamics with a Gauss law

General covariance and dynamics with a Gauss law

Shewhart control charts – A simple but not easy tool for data analysis

Shewhart control charts (ShCCs) are a powerful and technically simple tool for process variability analysis. However, simultaneously, they cannot be fully algorithmized and require deep process knowledge together with additional data analysis. ShCCs are well known, though, and the number of papers is great, as well as standards on ShCCs work in most countries, there are some serious obstacles for their effective application which are not being discussed in either educational or scientific literature. Just these problems are being considered in this paper. We analyzed two sides of standard assumption about data normality. First, we discuss the widely-spread misconception that measurement data are always distribu­ted according Gauss law. Then, it is shown how the deviation from normality may impact the method of ShCCs’ constructing and interpreting. Using a specific process data, we debate on right and wrong ways to build ShCC. Further, the paper describes two new definitions of assignable causes of variation: not changing (I-type) and changing (X-type) the system. At the end, we discuss how the work with ShCCs should be organized effectively. It is outlined that creating and analyzing ShCCs is always a system question of interaction between the process and the person who tries to improve this process.

One-loop partition function of gravity with leaky boundary conditions

Leaky boundary conditions in asymptotically AdS spacetimes are relevant to discuss black hole evaporation and the evolution of the Page curve via the island formula. We explore the consequences of leaky boundary conditions on the one-loop partition function of gravity. We focus on JT gravity minimally coupled to a scalar field whose normalizable and non-normalizable modes are both turned on, allowing for leakiness through the AdS boundary. Classically, this yields a flux-balance law relating the scalar news to the time derivative of the mass. Semi-classically, we argue that the usual diffeomorphism-invariant measure is ill-defined, suggesting that the area-non-preserving diffeomorphisms are broken at one loop. We calculate the associated anomaly and its implication on the gravitational Gauss law. Finally, we generalize our arguments to higher dimensions and dS.

Hairy black holes, scalar charges and extended thermodynamics

We explore the use of the recently defined scalar charge which satisfies a Gauss law in stationary spacetimes, in the context of theories with a scalar potential. We find new conditions that this potential has to satisfy in order to allow for static, asymptotically-flat black-hole solutions with regular horizons and non-trivial scalar field. These conditions are equivalent to some of the known ‘no-hair’ theorems (such as Bekenstein’s). We study the extended thermodynamics of these systems, deriving a first law and a Smarr formula. As an example, we study the Anabalón–Oliva hairy black hole.

Fluid/p-form duality

In this study, we demonstrate that an inviscid fluid in a near-equilibrium state, when viewed in the Lagrangian picture in d+1 spacetime dimensions, can be reformulated as a (d−1)-form gauge theory. We construct a fluid/p-form dictionary and show that volume-preserving diffeomorphisms on the fluid side manifest as a U(1) gauge symmetry on the (p+1)-form gauge theory side. Intriguingly, Kelvin's circulation theorem and the mass continuity equation respectively appear as the Gauss law and the Bianchi identity on the gauge theory side. Furthermore, we show that at the level of the sources, the vortices in the fluid side correspond to the p-branes in the gauge theory side. We also consider fluid mechanics in the presence of boundaries and examine the boundary symmetries and corresponding charges from both the fluid and gauge theory perspectives.

ELECTROTHERMO-RHEOLOGICAL PERISTALSIS: INVESTIGATING VELOCITY SLIP AND MODIFIED DARCY’S POROSITY EFFECTS

This study examines velocity slip and modified Darcy's porosity in an incompressible Carreau material flowing through an inclined channel under the influence of electroosmotic peristalsis. Modified Darcy's resistance, Gauss's law, and Carreau model equations are utilized in the fundamental equations of motion, Poisson's equation, and heat transfer equation. The objectives and methodology of the study are specified in order to effectively discuss the model results. The governing equations are susceptible to long wave and Debye-Huckle approximations. The non-dimensionalized equations identify controlling variables that facilitate the detection of temperature, velocity, and pressure gradients. Mathematica is used to solve the resultant nonlinear problem in order to analyze the variation in physical quantities of interest and temperature in the Carreau fluid flow pattern. The investigation provides primary findings, including quantitative statistics regarding velocity slip, modified Darcy's porosity, and temperature distribution. The results of the nonlinear system are graphically analyzed and discussed. It is noticed that higher values of electroosmotic parameter cause a decrease in temperature profiles, while a rise in Darcy's number causes a rise in the axial velocity's magnitude. Understanding the behaviors of Carreau fluids under the influence of electroosmotic peristalsis has potential applications in a vast array of biological microfluidic devices.

Analysis of clamped-clamped piezoelectric energy harvester under vortex induced vibration considering the stretching effect

This paper investigates the middle layer stretching effect on the dynamic behavior of an energy harvester clamped-clamped beam. Two foam cylinders that are attached together as a dumbbell are mounted on the middle part of the beam for vortex-induced vibrations. The piezoelectric patch collects the electrical energy produced by vortex induced vibration. In this analysis the coupled differential equations governing on the structure oscillation, harvested voltage and fluid lift force are established applying the Hamilton’s principle, Gauss law and wake oscillator model. The obtained differential equations are discretized using Galerkin method, and solved by both numerical and analytical perturbation methods. The results demonstrate that middle layer stretching effect has a significant effect on the lock-in domain, and the output electric voltage must be evaluated by considering this effect. It has been shown that for large values of cylinder diameter the difference between numerical and theoretical results increases due to increasing the middle layer stretching effect.