Piezoelectric materials exhibit coupling between their mechanical and electrical properties that is due to asymmetry in the underlying crystalline structure. These materials are anisotropic when considered at a macroscopic, continuum scale: their electrical and mechanical properties depend on direction. For these reasons it is crucial that we develop a mathematical description that is rich enough to characterize how the mechanical and electrical field variables transform with a change in coordinates. Section 2.1 describes how representations of vectors change when we vary the choice of basis. The transformation equations in this section are applicable to physical observables that are represented by vectors such as electric field, electric displacement, polarization, position, velocity, and acceleration. Section 2.2 generalizes this analysis and derives the transformation laws for nth order tensors . The transformation laws in Section 2.2 are applicable to quantities such as the stress tensor, linear strain tensor, and the higher order tensors that appear in the linear piezoelectric constitutive laws in Chapter 5. Section 2.3 discusses how symmetry properties are described in terms of invariance under transformations, and how symmetry considerations manifest in tensor invariance.

General theory of elasticity treats the anisotropic behaviour of media i.e. that property-dependence on spatial direction is taken care of. Examples of elastic media are rocks, building- and biological materials. The tensor concept is the most fundamental concept in the description of elastic anisotropy. Although a tensor describes a physical property and as such is independent of coordinate systems, the tensor can be represented by components referred to a coordinate system. A vector - which is a first rank tensor - is the most familiar quantity where the components are dependent on the coordinate system. The set of components is a representation of the vector. A scalar quantity like the length of a vector is independent of the coordinate system to which the vector is referred to. A main subject in my thesis, is a representation of the anisotropic elastic tensor by means of coordinate-free - or invariant - quantities to describe symmetry properties of the medium. A question in elastic anisotropy is how material symmetry can be determined from the components of the elastic tensor. This is of main concern in my thesis. In an arbitrary coordinate system an elastic tensor has 21 non-zero components. The exception is the isotropic tensor with some vanishing components in all coordinate systems and only two independent components. For ideal media with specified symmetry, there exist coordinate systems where some of the components are zero. How can a coordinate system which reduces

Most physical properties of crystals are defined by the relationship between two or more tensors and are therefore themselves represented by tensors. If a crystal is subjected to an influence presented by a tensor \( {I_{{j_1}}} \ldots {j_n} \), which produces a physical effect \( {E_{{i_1}}} \ldots {i_m} \), then a linear relationship between the influence and the effect is given by $$ {E_{{i_1}}} \ldots {i_m} = {m_{{i_1}}} \ldots {i_m}{j_1} \ldots {j_n}{I_{{j_1}}} \ldots {j_n} $$ (4.1) where the tensors E and I are called physical (field) tensors, and m is the material tensor. Note that, in higher-order effects, we may have more than one influence, i.e., $$ E_{i_1 } \ldots i_m = m_{i_1 } \ldots i_m ,j_1 \ldots j_n ,k_1 \ldots k_r I'_{j1} \ldots j_n I''_{k1} \ldots k_r $$ (4.2) as, say, in the piezomagnetoelectrism e ij = A ijkr P k M r , where e is the strain (or stress) tensor and P and M are the electric and magnetization vectors polarization, respectively.

In this chapter we introduce the tensor description of physical properties along with Neumann’s Principle relating symmetry to physical properties. As pointed out in the introduction, many different types of anisotropic properties are described in this book, but all have one thing in common: a physical property is a relationship between two measured quantities. Four examples are illustrated in Fig. 5.1. Elasticity is one of the standard equilibrium properties treated in crystal physics courses. The elastic compliance coefficients relate mechanical strain, the dependent variable, to mechanical stress, the independent variable. For small stresses and strains, the relationship is linear, but higher order elastic constants are needed to describe the departures from Hooke’s Law. Thermal conductivity is typical of the many transport properties in which a gradient leads to flow. Here the dependent variable is heat flow and the independent variable is a temperature gradient. Again the relationship is linear for small temperature gradients. Hysteretic materials such as ferromagnetic iron exhibit more complex physical properties involving domain wall motion. In this case magnetization is the dependent variable responsive to an applied magnetic field. The resulting magnetic susceptibility depends on the past history of the material. If the sample is initially unmagnetized, the magnetization will often involve only reversible domain wall motion for small magnetic fields. In this case the

It is assumed in the theory of relativity that physical quantities are represented by expressions derived from tensor components, and that the laws of nature may be expressed as equations stating the equality of two tensors. This assumption is made so as to satisfy the requirement that physical laws must be expressible in a form independent of the particular coordiinates used. If we start with this latter assumptioni, we are tempted to require merely that the equations expressing a law of nature be invariant, as a set, under transformations of coordinates, and the question arises as to the relation of equations of this type, referred to in the sequel as an invariant set of equations, to the tensor equations usually assumed. The requirement of invariance implies that there is a law of transformation for the equations in terms of the tranisformation of coordinates, which will be given if the equations involve merely teilsor components and the coordinates. If these quantities enter inlto the equations in a sufficiently simple manner (which, however, is as general as is required in most of the equations of physics), we may completely answer the question raised above by the theorem An invariant set of equations 'whose members are formed fromi the components of one or mtore tensors and point functions by addition, multiplication, and differentiation wvith respect to the coordinates is equivalent to a set of tensor equations. Here, as throughout this paper, the tensors relate to a R

In July 1925, exactly one century ago, Werner Heisenberg created Quantum Mechanics (QM) as an invariant-operational formalism by taking as a standpoint the intensive patterns that had been already observed by experimentalists in the lab. One of the few drawbacks of this proposal was the impossibility to conceptualize it in classical terms. Attempting to restore a somewhat classical spatiotemporal representation, six months later, Erwin Schrödinger would present a formalism grounded on a wave equation in configuration space. Taking as a standpoint Schrödinger’s wave mechanics together with the methodological guide of Bohr and logical positivists, Dirac would present in 1930 an axiomatic re-formulation of the theory of quanta that would become known as the “standard” version of QM (SQM). This new version of the theory, that is still today taught in physics classrooms all around the world, would become accepted regardless of its inconsistent narrative as a “recipe” intended (but unable) to predict (binary) measurement outcomes. After pointing out some of the many problems of SQM, in this work we propose to go back to Heisenberg’s matrix mechanics (and also to his understanding of physical theories). We will argue that his matrix mechanics, as an operationally-invariant standpoint, opens the door to a tensorial enlargement of the mathematical formalism capable to account for new phenomena.

We introduce the elastic fluctuation tensor to quantify the stochastic fluctuation of the apparent stiffness of finite microstructural volume elements. Typically, in computational homogenization using volume elements of finite size, the apparent stiffness converges to the effective stiffness as the volume element size tends to infinity, such that the material can be approximated as homogeneous on the macroscale. For volume elements of finite size, the apparent stiffness fluctuates on the macroscale. In thermal conductivity homogenization, the fluctuations can be quantified using the fourth-order fluctuation tensor, which computes as the infinite-volume limit of the apparent conductivity covariance, rescaled with the volume. The fluctuation tensor for linear elasticity is of tensor order eight. We show that this fluctuation tensor inherits the symmetry of its ensemble. For instance, rotational statistical symmetry of the ensemble leads to isotropy of the elastic fluctuation tensor. Using results from group representation theory, we define efficient representations of the eighth-order fluctuation tensor for various microstructure symmetry classes and discuss the physical meaning of individual components for the statistically isotropic case. We furthermore leverage symmetry to mitigate numerical errors, thereby reducing the expense of computing the fluctuation tensor. As an example material, we consider polypropylene reinforced by fibers and spherical inclusions. Additionally, we examine polycrystalline copper microstructures. By numerically computing the elastic fluctuation tensor, we confirm theoretical asymptotic convergence rates and symmetry properties. For many of the considered statistically isotropic microstructures, the fluctuations of isotropic stiffness components, which are often the only fluctuations reported, are negligible compared to isotropic fluctuations of the anisotropic stiffness components. Therefore, the full fluctuation tensor must be considered when quantifying the uncertainty of stochastic homogenization.

A calculation method for higher-order moments of physical quantities, including magnetization and energy, based on the higher-order tensor renormalization group is proposed. The physical observables are represented by impurity tensors. A systematic summation scheme provides coarse-grained tensors including multiple impurities. Our method is compared with the Monte Carlo method on the two-dimensional Potts model. While the nature of the transition of the $q$-state Potts model has been known for a long time owing to the analytical arguments, a clear numerical confirmation has been difficult due to extremely long correlation length in the weakly first-order transitions, e.g., for $q=5$. A jump of the Binder ratio precisely determines the transition temperature. The finite-size scaling analysis provides critical exponents and distinguishes the weakly first-order and the continuous transitions.

This book approaches condensed matter physics from the perspective of quantum information science, focusing on systems with strong interaction and unconventional order for which the usual condensed matter methods like the Landau paradigm or the free fermion framework break down. Concepts and tools in quantum information science such as entanglement, quantum circuits, and the tensor network representation prove to be highly useful in studying such systems. The goal of this book is to introduce these techniques and show how they lead to a new systematic way of characterizing and classifying quantum phases in condensed matter systems. The first part of the book introduces some basic concepts in quantum information theory which are then used to study the central topic explained in Part II: local Hamiltonians and their ground states. Part III focuses on one of the major new phenomena in strongly interacting systems, the topological order, and shows how it can essentially be defined and characterized in terms of entanglement. Part IV shows that the key entanglement structure of topological states can be captured using the tensor network representation, which provides a powerful tool in the classification of quantum phases. Finally, Part V discusses the exciting prospect at the intersection of quantum information and condensed matter physics – the unification of information and matter. Intended for graduate students and researchers in condensed matter physics, quantum information sc

product of two polar vectors are pseudovectors. A number of vector physical quantities behave as pseudovectors rather than polar vectors, including magnetic In physics and mathematics, a pseudovector (or axial vector) is a quantity that transforms like a vector under continuous rigid transformations such as rotations or translations, but which does not transform like a vector under certain discontinuous rigid transformations such as reflections. For example, the angular velocity of a rotating object is a pseudovector because, when the object is reflec In… The definition of a "vector" in physics (including both polar vectors and pseudovectors) is more specific than the mathematical definition of "vector" (namely, any element of an abstract vector space). Under the physics definition, a "vector" is required to have components that "transform" in a certain way under a proper rotation: In particular, if everything in the universe were rotated, the vector would rotate in exactly the same way. (The coordinate system is fixed in this discussion; in other words this is the perspective of active transformations.) Mathematically, if everything in the universe undergoes a rotation described by a rotation matrix R, so that a displacement vector x is transformed to x′ = Rx, then any "vector" v must be similarly transformed to v′ = Rv. This important requirement is what distinguishes a vector (which might be composed of, for example, the x-, y-, and z-components of velocity) from any other triplet of physical quantities (For example, the length, width, and height of a rectangular box cannot be considered the three components of a vector, since rotating the box does not appropriately transform these three components.) (In the language of differential geometry, this requirement is equivalent to defining a vector to be a tensor of contravariant rank one. In this more general framework, higher rank tensors can also have arbitrarily many and mixed covariant and contravariant ranks at the same time, denoted by raised and lowered indices within the Einstein summation convention.) A basic and rather concrete example is that of row and column vectors under the usual matrix multiplication operator: in one order they yield the dot product, which is just a scalar and as such a rank zero tensor, while in the other they yield the dyadic product, which is a matrix representing a rank two mixed tensor, with one contravariant and one covariant index. As such, the noncommutativity of standard matrix algebra can be used to keep track of the distinction between covariant and contravariant vectors. This is in fact how the bookkeeping was done before the more formal and generalised tensor notation came to be. It still manifests itself in how the…

Microstructure-sensitive prediction of elastoplastic response remains a recurring bottleneck in multiscale damage and fatigue modeling, where large ensembles of statistically distinct polycrystals are required to quantify variability and extreme-value behavior. In this work, we develop a multitask graph neural network (GNN) surrogate that maps dual-phase ferrite–martensite polycrystal microstructures to Statistical Volume Element (SVE)-level elastoplastic Quantities of Interest (QoIs). Each SVE is represented as a grain-adjacency graph, with node features encoding phase, geometry, and crystallographic orientation, and edge features encoding relative misorientation. A message-passing graph convolution generates node embeddings, which are pooled into a graph representation and passed to a multitask regression head that jointly predicts 10 scalar QoIs and vector-valued stress–strain responses in orthogonal loading directions across multiple martensite volume fractions and SVE sizes. Results show high accuracy for scalar QoIs and strong agreement for full stress–strain trajectories, with population envelopes reproducing both median behavior and finite-SVE variability across compositions and partition scales. A unified model trained on pooled volume-fraction data preserves most within-regime accuracy relative to regime-specific models while also capturing the broader cross-regime variation reflected in the pooled test set. Distributional comparisons further demonstrate that the su

Parton distributions encode the momentum-space structure and, in their generalizations, the spatial tomography of quarks and gluons inside hadrons, the building blocks of visible matter. We present a unified neural-network approach that learns these distributions directly from matrix elements calculated via numerical simulations of quantum chromodynamics (QCD) on the lattice by fitting two complementary inputs simultaneously: data matched to physical quantities via known momentum-space and coordinate-space formalisms. Utilizing data from both methods stabilizes the extraction and mitigates biases that can arise when either is used alone. We validate the method on controlled mock data and apply it to lattice-QCD matrix elements to extract parton distribution functions (PDFs). We show benefits of such an approach for determining the physical quantities. We further extend the framework to zero-skewness generalized parton distributions and demonstrate nucleon tomography within the same neural-network parameterization. Our results provide an adaptable and systematically improvable approach for extracting partonic distributions from Euclidean correlators. It can incorporate polarization, additional channels, and future experimental constraints from current and future facilities, such as the Electron-Ion Collider.

two of these conditions are naturally taken from Poisson's equation. Since it may be proved mathematically that all such differential tensors can be formed