Tensor Quantities Research Articles

The Upper Crustal Deformation Field of Greece Inferred From GPS Data and Its Correlation With Earthquake Occurrence

AbstractWe present a new geodetic strain rate and rotation rate model for Greece that has been derived using a highly dense GPS velocity field. The spatial distribution and the resolved rates of the various velocity gradient tensor quantities provided updated constraints on the present‐day upper crustal deformation in the region and revealed new details not reported previously. The spatial distribution of the second invariant demonstrated that the overall magnitude of strain rates is highest across two well‐defined provinces. The first follows the North Anatolian Fault and its two branches within the north Aegean, crosses central Greece and through the Gulf of Corinth it terminates in western Greece, while the second encompasses the extensional province of western Turkey and the eastern Aegean Sea islands. Our estimates revealed that shearing affects some of the fault‐bounded grabens of central Greece that lie to the SW of the North Aegean Basin implying considerable oblique extension. We identified a narrow region of counterclockwise rotation whose location and kinematics have been induced by the net effect across the intersection of the clockwise rotating domains of western and central Greece. The Aegean microplate and the Anatolian plate are separated by a wide transition zone which accommodates the curved stretching of the entire plate system. In both edges of the Hellenic forearc the dominant mode of crustal strain is E‐W extension. We found that earthquakes of M ≥ 5.6 are spatially well‐correlated with high‐strain areas, indicating that strain rate mapping could be used to inform future probabilistic seismic hazard analyses.

A theory of magnetoresistance of non-magnetic metal on magnon valves

One recent exciting development in the field of magnonics is the discovery of universal unusual anisotropic magnetoresistance (UAMR) in nanometer-thick non-magnetic (NM) metallic bars that are deposited on magnon valves of two ferromagnetic insulators (FIs) sandwiching an NM metal. This UAMR has the same angular dependencies as various bilayers consisting of at least one magnetic layer and at least one metallic layer. This suggests that the UAMR of different systems may originate from the same physics, which is yet to be fully understood. Here, we reveal the common feature shared by all these systems: two-vector dependencies of tensor quantities. Specifically, the resistivity of an NM metallic bar depends on the magnetization of its adjacent FI due to the quantum penetration of itinerant electrons of the metallic bar into the FI and on a perpendicular field at the interface of the bar and the FI. We demonstrate that the two-vector dependence of the resistivity tensor is responsible for the observed universal UAMR of metallic bars on magnon valves, independent of the details of the microscopic interactions in different materials. We also propose experiments that can test this theory.

Eulerian finite element implementations of a dislocation density-based continuum model

In Eulerian finite element simulations, the mesh moves relative to the material. After every change of position between the mesh and the material, the state variables are interpolated to the new mesh position, which is referred to as advection. Large strain crystal plasticity models are based on the multiplicative decomposition of the total deformation gradient. The stress is evaluated as a function of the thermoelastic strain, temperature, and other state variables. Advection of tensor quantities, such as the strain, is coupled with possibly significant advection errors. In an effort to reduce the advection errors, we develop two rate forms of an established dislocation density-based continuum model. To that end, we replace the multiplicative decomposition of the deformation gradient with the additive decomposition of the velocity gradient, and define the stress rate instead of the total stress. The Eulerian implementation is compared with Lagrangian calculations, and two numerical examples with severe deformation levels are presented.

Tensors and Algebras: An Algebraic Spacetime Interpretation for Tensor Models

The quest for a consistent theory for quantum gravity is one of the most challenging problems in theoretical high-energy physics. An often-used approach is to describe the gravitational degrees of freedom by the metric tensor or related variables, and finding a way to quantise this. In the canonical tensor model, the gravitational degrees of freedom are encoded in a tensorial quantity $P_{abc}$, and this quantity is subsequently quantised. This makes the quantisation much more straightforward mathematically, but the interpretation of this tensor as a spacetime is less evident. In this work we take a first step towards fully understanding the relationship to spacetime. By considering $P_{abc}$ as the generator of an algebra of functions, we first describe how we can recover the topology and the measure of a compact Riemannian manifold. Using the tensor rank decomposition, we then generalise this principle in order to have a well-defined notion of the topology and geometry for a large class of tensors $P_{abc}$. We provide some examples of the emergence of a topology and measure of both exact and perturbed Riemannian manifolds, and of a purely algebraically-defined space called the semi-local circle.

Top-down metacomputing with algebraic dimensionality raising for automating theory-building to enable directly computable multiphysics models

The need to address the fundamental issues associated with the computational model generation process in the context of multiphysics theories that observe conservation laws and relevant assumptions motivates the present work. Accordingly, the goal of this paper is to describe, discuss, and demonstrate the benefits and implications of using metacomputing (i.e., the utilization of computational technologies for computing or generating programs that, in turn, compute the desired results) for theory or model generation, further assisted by metacomputing for raising the semantic and syntactic dimensionality of physics models for continua. The issues associated with the model generation process are identified at the outset. Then, a top-down approach that contains three distinct metacomputing activities is described. First, a meta-theoretical description of generating continuum multiphysics models defined in terms of quantities instantiated from the field of real numbers is described, along with its computational implementation. Then, a computational framework for raising the semantic dimensionality from the reals to the hypercomplex numbers is presented for both 2D and 3D continuum multiphysics problems. The semantic dimensionality is expressed in terms of the order of the algebra used to define the tensorial quantities of interest and their associated operations, as they participate in representing physics models expressed initially as systems of partial differential equations. The utilization of hypercomplex algebras, such as those for complex numbers and quaternions, is demonstrated for representative applications of models. Subsequently, syntactic dimensionality is defined in terms of the basis of the space utilized to write equational systems. Finally, a prototype of a computational system for converting equational systems to directed graphs termed “Algebraic Solution Graphs” is described. Examples of how to utilize these technologies are also presented to convey the efficacy and efficiency of the proposed approach.

Anisotropy of magnetic damping in Ta/CoFeB/MgO heterostructures

Magnetic damping controls the performance and operational speed of many spintronics devices. Being a tensor quantity, the damping in magnetic thin films often shows anisotropic behavior with the magnetization orientation. Here, we have studied the anisotropy of damping in Ta/CoFeB/MgO heterostructures, deposited on thermally oxidized Si substrates, as a function of the orientation of magnetization. By performing ferromagnetic resonance (FMR) measurements based on spin pumping and inverse spin Hall effect (ISHE), we extract the damping parameter in those films and find that the anisotropy of damping contains four-fold and two-fold anisotropy terms. We infer that four-fold anisotropy originates from two-magnon scattering (TMS). By studying reference Ta/CoFeB/MgO films, deposited on LiNbO3 substrates, we find that the two-fold anisotropy is correlated with in-plane magnetic anisotropy (IMA) of the films, suggesting its origin as the anisotropy in bulk spin–orbit coupling (SOC) of CoFeB film. We conclude that when IMA is very small, it’s correlation with two-fold anisotropy cannot be experimentally identified. However, as IMA increases, it starts to show a correlation with two-fold anisotropy in damping. These results will be beneficial for designing future spintronics devices.

An equivariant neural operator for developing nonlocal tensorial constitutive models

Developing robust constitutive models is a fundamental and longstanding problem for accelerating the simulation of multiscale physics. Machine learning provides promising tools to construct constitutive models based on various calibration data. However, with either traditional or machine learning techniques, the constitutive model for tensorial quantities has been much less studied than scalar quantities due to more complicated physics, even though the former plays an important role in many scientific and engineering applications. In this work, we propose a neural operator to develop nonlocal constitutive models for tensorial quantities through a vector-cloud neural network with equivariance (VCNN-e). The VCNN-e respects all the invariance properties desired by constitutive models, faithfully reflects the region of influence in physics, and is applicable to different spatial resolutions. By design, the model guarantees that the predicted tensor is invariant to the frame translation and ordering (permutation) of the neighboring points. Furthermore, it is equivariant to the frame rotation, i.e., the output tensor co-rotates with the coordinate frame. We evaluate the VCNN-e by using it to emulate the Reynolds stress transport model for turbulent flows, which directly computes the Reynolds stress tensor to close the Reynolds-averaged Navier–Stokes (RANS) equations. The evaluation is performed in two situations: (1) emulating the Reynolds stress model through synthetic data generated from the Reynolds stress transport equations with closure models, and (2) predicting the Reynolds stress by learning from data generated from direct numerical simulations. Such a priori evaluations of the proposed network on realistic and challenging datasets pave the way for developing and calibrating robust and nonlocal, non-equilibrium constitutive models for the RANS equations and other mechanical problems.

Custom Nonlinearity Profile for Integrated Quantum Light Sources

Heralded single-photon sources are a fundamental building block for optical quantum technologies. These sources need to be unfiltered and integrated to have good scalability and avoid unnecessary losses. To attain this goal, it is necessary to control the effective nonlinearity seen by the fields as they mix and propagate in a waveguide source. In this paper, we introduce a method to design nonlinear waveguides with arbitrarily shaped effective nonlinearity profiles. The method takes advantage of the fact that the second-order nonlinear response is a tensor quantity and thus the local effective nonlinearity of a material depends on the propagation direction of the fields participating in the interaction. Thus, by locally changing the propagation direction of the fields we can modulate the wave-mixing process. Our method allows for the waveguide fabrication process to be significantly simplified. The material structure of the waveguide is made by a single crystal, no longer needing oriented patterning or periodic poling. We use our method to design waveguides with a nonlinearity profile that is Gaussian in the propagation length, allowing the generation of perfectly pure heralded single photons.

Role of Strong Matter‐Field Coupling on Axial Complex Systems

AbstractEven though extended gravity theories have come up with different healthy measures of complexity for spherical and non‐spherical systems, the study of complexity factors for axially symmetric bodies under the influence of strong matter‐field coupling is still up for debate. The formation of complexity factors for static axially symmetric bodies is induced by the cracking of the Riemann curvature tensor in an orthogonal manner as initiated by Herrera et al. [Phys. Rev. D 99, 044049(2019)]. We provide the continuation of this work in the gravitational theory, where that indicates the powerful matter‐field coupling due to the bilateral influence of the curvature and matter tensors. We proceed our analysis by considering axial geometry confined with anisotropic fluid constituents. In this context, we compute the field equations for the system under discussion. To examine the source's energy content, two dynamical equations that correspond to independent coordinates are constructed. The curvature tensor is split into tensorial quantities to formulate the scalar functions (structure scalars), which involve the physical attributes of the source in the domain of high curvature regimes. Among all, three of the scalars 's are nominated as complexity factors, whose inclusion enhances the inhomogeneity of the propagating system. The vanishing complexity factors criteria are set out for the specific form of the cosmic model that provides the favorable setting to minimize the system's complexity and boost its stability. Furthermore, some analytical insights under various backgrounds of the matter spheroids are presented.

Cadabra and Python algorithms in general relativity and cosmology II: Gravitational waves

Computer Algebra Systems (CASs) like Cadabra Software play a prominent role in a wide range of research activities in physics and related fields. We show how Cadabra language is easily implemented in the well established Python programming framework, gaining excellent flexibility and customization to address the issue of tensor perturbations in General Relativity. We obtain a performing algorithm to decompose tensorial quantities up to any perturbative order of the metric. The features of our code are tested by discussing some concrete computational issues in research activities related to first/higher-order gravitational waves.

Analytical Modeling of Magnetic Field Distribution at No Load for Surface Mounted Permanent Magnet Machines

This paper presents an analytical study of the air-gap magnetic field of a Surface Permanent Magnet (SPM) linear machine under no load. By means of the method of images, the complex expression of the magnetic field of a conductor, inside the air gap between two smooth iron surfaces, is retrieved. Then, integrating the conductor expression, the formulation of the magnetic field of a current sheet and thus the one of a SPM, using two vertical current sheets, is obtained. At last, the no-load magnetic field expression, for a generic time instant, of a slotless machine is retrieved. The novelty of the proposed approach is the availability, due to a different calculation approach, of a unique closed-form formulation for the slotless machine air-gap field, a quantity that, in literature, is usually present in Fourier series formulation. Additionally, as a means to calculate integral quantities and show the goodness of the method a complex slotting function is introduced to account for the slotted geometry. Finally, starting from Lorenz’s force formulation, the expression of the Maxwell tensor in complex form is retrieved and the contribution of forces, integral of the complex stress tensor quantity, will be calculated and compared with FEM simulations, showing a good agreement also with the analytical slotted model.

Integrated intravoxel incoherent motion tensor and diffusion tensor brain MRI in a single fast acquisition.

The acquisition of intravoxel incoherent motion (IVIM) data and diffusion tensor imaging (DTI) data from the brain can be integrated into a single measurement, which offers the possibility to determine orientation-dependent (tensorial) perfusion parameters in addition to established IVIM and DTI parameters. The purpose of this study was to evaluate the feasibility of such a protocol with a clinically feasible scan time below 6 min and to use a model-selection approach to find a set of DTI and IVIM tensor parameters that most adequately describes the acquired data. Diffusion-weighted images of the brain were acquired at 3 T in 20 elderly participants with cerebral small vessel disease using a multiband echoplanar imaging sequence with 15 b-values between 0 and 1000 s/mm2 and six non-collinear diffusion gradient directions for each b-value. Seven different IVIM-diffusion models with 4 to 14 parameters were implemented, which modeled diffusion and pseudo-diffusion as scalar or tensor quantities. The models were compared with respect to their fitting performance based on the goodness of fit (sum of squared fit residuals, chi2 ) and their Akaike weights (calculated from the corrected Akaike information criterion). Lowest chi2 values were found using the model with the largest number of model parameters. However, significantly highest Akaike weights indicating the most appropriate models for the acquired data were found with a nine-parameter IVIM-DTI model (with isotropic perfusion modeling) in normal-appearing white matter (NAWM), and with an 11-parameter model (IVIM-DTI with additional pseudo-diffusion anisotropy) in white matter with hyperintensities (WMH) and in gray matter (GM). The latter model allowed for the additional calculation of the fractional anisotropy of the pseudo-diffusion tensor (with a median value of 0.45 in NAWM, 0.23 in WMH, and 0.36 in GM), which is not accessible with the usually performed IVIM acquisitions based on three orthogonal diffusion-gradient directions.

Ion-beam radiation-induced Eshelby transformations: The mean and variance in hydrostatic and shear residual stresses

Ion beam plays a pivotal role in ion implantations and the fabrication of nanostructures. However, there lacks a quantitative model to describe the residual stresses associated with the ion-beam radiation. Radiation-induced residual stress/transformation strain have been mostly recognized in the hydrostatic sub strain space. Here, we use molecular dynamics (MD) simulations to show that the response of a material to irradiation is generally anisotropic that depends on the ion-beam direction, and should be described using tensorial quantities. We demonstrate that accelerator-based ion beam irradiation, combined with the intrinsic lattice anisotropy and externally induced anisotropy (such as anisotropic mechanical loadings), causes radiation-actuated shear transformation strains in addition to hydrostatic expansion. We map out these complex correlations for several materials. Radiation-induced defects are shown to be responsible for residual shear stresses in the manner of Eshelby inclusion transformation. We propose such tensorial response model should be considered for accurate nanoscale fabrication using ion-beam irradiation.

‘Anti-commutable’ local pre-Leibniz algebroids and admissible connections

The concept of algebroid is convenient as a basis for constructions of geometrical frameworks. For example, metric-affine and generalized geometries can be written on Lie and Courant algebroids, respectively. Furthermore, string theories might make use of many other algebroids such as metric algebroids, higher Courant algebroids, or conformal Courant algebroids. Working on the possibly most general algebroid structure, which generalizes many of the algebroids used in the literature, is fruitful as it creates a chance to study all of them at once. Local pre-Leibniz algebroids are such general ones in which metric-connection geometries are possible to construct. On the other hand, the existence of the ‘locality operator’, which is present for the left-Leibniz rule for the bracket, necessitates the modification of torsion and curvature operators in order to achieve tensorial quantities. In this paper, this modification of torsion and curvature is explained from the point of view that the modification is applied to the bracket instead. This leads one to consider ‘anti-commutable’ local pre-Leibniz algebroids which satisfy an anti-commutativity-like property defined with respect to a choice of an equivalence class of connections. These ‘admissible’ connections are claimed to be the necessary ones while working on a geometry of algebroids. This claim is due to the fact that one can prove many desirable properties and relations if one uses only admissible connections. For instance, for admissible connections, we prove the first and second Bianchi identities, Cartan structure equations, Cartan magic formula, the construction of Levi-Civita connections, the decomposition of connection in terms of torsion and non-metricity. These all are possible because the modified bracket becomes anti-symmetric for an admissible connection so that one can apply the machinery of almost- or pre-Lie algebroids. We investigate various algebroid structures from the literature and show that they admit admissible connections which are metric-compatible in some generalized sense. Moreover, we prove that local pre-Leibniz algebroids that are not anti-commutable cannot be equipped with a torsion-free, and in particular Levi-Civita, connection.

The gravitational energy-momentum pseudo-tensor in higher-order theories of gravity

The problem of non-localizability and the non-uniqueness of gravitational energy in general relativity has been considered by many authors. Several gravitational pseudo-tensor prescriptions have been proposed by physicists, such as Einstein, Tolman, Landau, Lifshitz, Papapetrou, Moller, andWeinberg. We examine here the energy-momentum complex in higher-order theories of gravity applying the Noether theorem for the invariance of gravitational action under rigid translations. This, in general, is not a tensor quantity because it is not a covariant object but only an affine tensor, that is, a pseudo-tensor. Therefore we propose a possible prescription of gravitational energy and momentum density for ?k gravity governed by the gravitational Lagrangian L1 = (R + a0R2 + Pp k=1 akR?kR) ??g and generally for n-order gravity described by the gravitational Lagrangian L = L (g??, g??,i1, 1??,i1i2, g??,i1i2i3 ,..., g??,i1i2i3...in). The extended pseudo-tensor reduces to the one introduced by Einstein in the limit of general relativity where corrections vanish. Then, we explicitly show a useful calculation, i.e., the power per unit solid angle ? emitted by a massive system and carried by a gravitational wave in the direction ? x for a fixed wave number k. We fix a suitable gauge, by means of the average value of the pseudo-tensor over a spacetime domain and we verify that the local pseudo-tensor conservation holds. The gravitational energy-momentum pseudo-tensor may be a useful tool to search for possible further gravitational modes beyond the two standard ones of general relativity. Their finding could be a possible observable signatures for alternative theories of gravity.

Gateway-like absurdly benign traversable wormhole solutions

A class of wormhole solutions is constructed that has restricted polar degrees of freedom to achieve a gateway-like configuration. This compels the use of distribution-valued metrics and connections which further compels the use of neutrix product of distributions, to define distribution-valued curvature, Einstein tensor, and other relevant quantities. The solution demands a spacetime with non-Riemannian effects like non-metricity to be consistent and well-defined, due to the non-associativity of the neutrix product. Finally, the ideal gateway configuration where the negative energy requirement is zero is derived.

Atomistic neural network representations for chemical dynamics simulations of molecular, condensed phase, and interfacial systems: Efficiency, representability, and generalization

AbstractMachine learning techniques have been widely applied in many fields of chemistry, physics, biology, and materials science. One of the most fruitful applications is machine learning of the complicated multidimensional function of potential energy or related electronic properties from discrete quantum chemical data. In particular, substantial efforts have been dedicated to developing various atomistic neural network (AtNN) representations, which refer to a family of methods expressing the targeted physical quantity as a sum of atomic components represented by atomic NNs. This class of approaches not only fully preserves the physical symmetry of the system but also scales linearly with respect to the size of a system, enabling accurate and efficient chemical dynamics and spectroscopic simulations in complicated systems and even a number of variably sized systems across the phases. In this review, we discuss different strategies in developing highly efficient and representable AtNN potentials, and in generalizing these scalar AtNN models to learn vectorial and tensorial quantities with the correct rotational equivariance. We also review active learning algorithms to generate practical AtNN models and present selected examples of AtNN applications in gas‐surface systems to demonstrate their capabilities of accurately representing both molecular systems and condensed phase systems. We conclude this review by pointing out remaining challenges for the further development of more reliable, transferable, and scalable AtNN representations in more application scenarios.This article is categorized under:Data Science > Artificial Intelligence/Machine LearningMolecular and Statistical Mechanics > Molecular Interactions

Continuum mechanics modeling of complex fluid systems following Oldroyd's seminal 1950 work

A review of some key developments in the continuum mechanics – based macroscopic modelling of complex fluid flows, following the pioneering work of Oldroyd in the 1950s will be presented. We start by reviewing the major developments achieved by Oldroyd, namely how his work established rules for consistency in developing continuum stress constitutive models extending the material into objective time derivatives for tensor quantities, with first application the stress tensor in viscoelastic constitutive relationships. We then show how the impact of his work to viscoelastic fluid flows has been amplified from the kinetic theory work of Bird and others who established a firm macromolecular structural connection. The use of objective time derivatives was then extended to include second order conformation tensors and through them the connection was made to internal deformation energy by Marrucci. We then further demonstrate the impact of this work by analysing the models through the microstructural descriptions offered within the non-equilibrium thermodynamics formalism. We then conclude with a brief mentioning of direct applications of the original Oldroyd-B model (through stability analyses, etc.) towards bettering our understanding of the rheological behaviour of complex fluid systems.

Ion-scale Transition of Plasma Turbulence: Pressure–Strain Effect

We investigate properties of solar-wind-like plasma turbulence using direct numerical simulations. We analyze the transition from large, magnetohydrodynamic (MHD) scales to the ion characteristic ones using two-dimensional hybrid (fluid electrons and kinetic ions) simulations. To capture and quantify turbulence properties, we apply the Karman–Howarth–Monin (KHM) equation for compressible Hall–MHD (extended by considering the plasma pressure as a tensor quantity) to the numerical results. The KHM analysis indicates that the transition from MHD to ion scales (the so-called ion break in the power spectrum) results from a combination of an onset of Hall physics and an effective dissipation owing to the pressure–strain energy-exchange channel and resistivity. We discuss the simulation results in the context of the solar wind.

Predicting tensorial molecular properties with equivariant machine learning models

Embedding molecular symmetries into machine-learning models is key for efficient learning of chemico-physical scalar properties, but little evidence on how to extend the same strategy to tensorial quantities exists. Here we formulate a scalable equivariant machine-learning model based on local atomic environment descriptors. We apply it to a series of molecules and show that accurate predictions can be achieved for a comprehensive list of dielectric and magnetic tensorial properties of different ranks. These results show that equivariant models are a promising platform to extend the scope of machine learning in materials modelling.