Electricity at the macroscale and its microscopic origins

https://doi.org/10.48550/arXiv.2403.13981

Abstract

This work examines electrical structures, and the relationships between electrical structures at the microscale and electrical structures at the macroscale. By structures I mean both physical structures, such as spatial distributions of charge and potential, and the mathematical structures used to specify physical structures and to relate them to one another. I do not discuss magnetism and what little I say about energetics is incidental.

I define the fields that specify electrical macrostructure, and their rates of change, in terms of the microscopic charge density $\rho$, electric field $\me$, electric potential $\phi$, and their rates of change. To deduce these definitions, I begin laying new foundations of a general theory of structure homogenization, meaning a theory of how any observable macroscopic field $\Nu$ is related to spatial averages of its microscopic counterpart $\nu$. An integral part of structure homogenization theory is the definition of macroscopic excess fields in terms of microscopic fields. The excess field of ${\Nu:\realone^n\to\realone}$ on the boundary ${\partial\manifold}$ of a finite-measure subset $\manifold$ of $\realone^n$ is the field ${\bsigmaNu:\partial\manifold\to\realone}$ to which it is related by the generalized Stokes theorem, ${\int_\manifold \Nu\dd{\volumeform}=\int_{\partial\manifold}\bsigmaNu\volumeform}$; where ${\volumeform}$ and ${\dd{\volumeform}}$ are volume forms on ${\partial\manifold}$ and ${\manifold}$, respectively, and ${\Nu\dd{\volumeform}\equiv\dd{\left(\bsigmaNu\volumeform\right)}}$. For example, the macroscopic volumetric charge density $\Rho$ in a material $\manifold$ is related to the areal charge density $\bsigma$ on its surface by ${\int_{\manifold} \Rho\ddpow{3}{\br} = \int_{\partial\manifold}\bsigma\ddpow{2}{\bs}}$ and by ${\Rho\ddpow{3}{\br}\equiv\dd{\left(\bsigma\ddpow{2}{\bs}\right)}}$. I derive an expression for ${\bsigmaNu[\nu]}$, which generalizes Finnis’s expression [Finnis, 1998] for excess fields at the surfaces of crystals (e.g., surface charge density ${\bsigma[\rho]}$) to disordered microstructures.

I use homogenization theory to define the macroscopic potential ${\bphi\equiv\bphi[\phi]}$, electric field ${\E\equiv\E[\me]}$, and charge density ${\Rho\equiv\Rho[\rho]}$, and I define the macroscopic current density as ${\J\equiv \dbsigma[\drho]}$. Using the microscopic theory, or vacuum theory, of electromagnetism as my starting point, I deduce that the relationships between these macroscopic fields are identical in form to the relationships between their microscopic counterparts. Without invoking quantum mechanics, I use the definitions ${\J\equiv\dbsigma}$ and ${\bsigma\equiv\bsigma[\rho]}$ to derive the expressions for so-called polarization current established by the Modern Theory of Polarization (MTOP). I prove that the bulk-average electric potential, or mean inner potential, $\bphi$, vanishes in a macroscopically-uniform charge-neutral material, and I show that when a crystal lattice lacks inversion symmetry, it does not imply the existence of macroscopic $\E$ or $\pp$ fields in the crystal’s bulk.

I point out that symmetry is scale-dependent. Therefore, if anisotropy of the microstructure does not manifest as anisotropy of the macrostructure, it cannot be the origin of a macroscopic vector field. Only anisotropy of the macrostructure can bestow directionality at the macroscale. The macroscopic charge density ${\Rho}$ is isotropic in the bulks of most materials, because it vanishes at every point. This implies that, regardless of the microstructure $\rho$, a macroscopic electric field cannot emanate from the bulk. I find that all relationships between observable macroscopic fields can be expressed mathematically without introducing the polarization ($\pp$) and electric displacement ($\D$) fields, neither of which is observable. Arguments for the existence of $\pp$ and $\D$, and interpretations of them, have varied since they were introduced in the 19th century. I argue that none of these arguments and interpretations are valid, and that macroscale isotropy prohibits the existence of $\pp$ and $\D$ fields.

Single-particle statistical states play prominent roles in the MTOP and in most textbook descriptions of electrical microstructures. Therefore I discuss several kinds of $1$-particle states in a many-particle system. I derive exact expressions for the energy of a set of interacting indistinguishable particles in a pure state in terms of the state’s natural orbitals, which are the eigenfunctions of its $1$-particle density matrix. These expressions strengthen an already-strong case against the traditional idea that a material’s electron number density has a substructure of localized orbitals with almost integral occupancies.

I do not invoke quantum mechanics axiomatically in this work. Instead I point out that statistical states of classical dynamical systems can be specified by wavefunctions and density matrices that have the same basic properties as their quantum mechanical counterparts; and I include or append classical derivations of all aspects of quantum mechanics that are relevant to this work.