Electricity at the macroscale and its microscopic origins

TL;DR

This work argues that macroscopic electricity can be consistently described as spatial averages of microscopic fields without invoking the unobservable polarization ${\mathbf{P}}$ or displacement ${\mathbf{D}}$ fields. It develops a rigorous homogenization framework that yields macrofields and macroscopic excesses (e.g., surface charge) from microstructure, and shows that the macroscopic electric field in a charge-neutral bulk vanishes, with changes arising at interfaces. The Modern Theory of Polarization is reinterpreted as a special case within a broader classical-statistical framework, and the polarization current ${\mathbf{J}^{(p)}}$ is derived from evolving bulk charge densities without requiring quantum-mechanical constructs. The results challenge long-standing notions such as Bethe's mean inner potential and Lorentz's local-field ideas, and emphasize scale-dependent symmetry and the primacy of surface/excess fields in determining macroscopic observables. Overall, the paper provides a comprehensive, modular foundation for linking microstructure to macroelectrostatics, with implications for interpreting experiments and simulations across condensed matter, materials science, and nanostructures.

Abstract

I define the fields that describe electrical macrostructure, and their rates of change, in terms of the microscopic charge density, electric field, electric potential, and their rates of change. To deduce these definitions, I lay some new foundations of a theory of how observable macroscopic fields are related to spatial averages of their microscopic counterparts. I find that the relationships between macroscopic fields are identical in form to the relationships between their microscopic counterparts, meaning that the $\vec{P}$ and ${\vec{D}}$ fields do not appear in them. Without invoking quantum mechanics, I derive the expressions for polarization current established by the Modern Theory of Polarization. I prove that the bulk-average electric potential, or mean inner potential, 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 $\vec{E}$ or $\vec{P}$ 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. The macroscopic charge density vanishes in a material's bulk. Therefore, regardless of the microstructure, a macroscopic $\vec{E}$ field cannot emanate from the bulk. I find that all relationships between observable macroscopic fields can be expressed mathematically without introducing the polarization ($\vec{P}$) and electric displacement ($\vec{D}$) fields, neither of which is observable. I also show that most `quantum mechanical' aspects of the existing microscopic theory of electricity in materials are compatible with, or required features of, a statistical theory of classical particles whose charges and masses are comparable to those of electrons and nuclei.

Figures (22)

• Figure 1: Schematic of a quasi one dimensional material which is macroscopically uniform but microscopically non-uniform. See Fig. \ref{['fig:crystal_dipole']} for a schematic of a three dimensional polarized charge-neutral material.
• Figure 2: Charge density $\rho$ as function of position $x$ in three one dimensional crystals. The crystals in (a) and (b) lack inversion symmetry, but the crystal in (c) has inversion symmetry, with two inversion centers per primitive cell $\Omega$. In (c) a choice of primitive unit cell whose dipole moment is zero is outlined in green. If $\rho(x)$ changes continuously and uniformly between the densities plotted in red and blue, a macroscopic current flows in crystals (a) and (b) because the probability that the net movement of charge in direction $\hat{x}$ equals the net movement of charge in the inequivalent direction ${-\hat{x}}$ is zero. In (c) the symmetry of the crystal forbids a macroscopic flow of charge because the net movement of charge relative to an inversion center cannot differ between the two equivalent directions $\hat{x}$ and $-\hat{x}$. One way to see this is to note that in (c) most of space can be tiled with unit cells $\Omega$ whose dipole moment remains zero throughout the changing of the density. There remains only the two shaded regions of combined width ${a=\abs{\Omega}}$ at the left and right boundaries of the chunk of bulk crystal comprised of ${M_{\Omega}=6}$ primitive cells. In the limit of large ${M_{\Omega}}$ the change in the distance between $x_b$ and the center of charge of the $M_{\Omega}$ cells, divided by their combined width ${M_{\Omega} a}$, vanishes. In (a) and (b) the current cannot vanish because the ${\hat{x}}$ and ${-\hat{x}}$ directions are inequivalent. An important question, which the MTOP solved for quantum systems, is how the current can be calculated from an evolving bulk microstructure, i.e., without knowing or calculating how much charge accumulates at surfaces. If the integrals $q_1$ and $q_2$ of the two peaks per unit cell in (a) remain constant, the current per unit length is simply ${\left(q_1\dot{x}_1+q_2\dot{x}_2\right)/a}$, where ${\dot{x}_1}$ and ${\dot{x}_2}$ are the velocities with which the peaks move. However if the charge density is not organized into packets of fixed charge, as in (b), the definition of current is much less obvious.
• Figure 3: Schematic of a polarized three dimensional material, which complements the schematic of the quasi-one dimensional material in Fig. \ref{['fig:material']}. The material is charge neutral, and its bulk (shaded green) is macroscopically charge neutral. The material has a finite dipole moment, ${\mathbfcal{D}\hat{x}}$, and on the macroscale the surfaces with outward normals ${\hat{x}}$ and ${-\hat{x}}$ carry equal and opposite areal charge densities, ${\boldsymbol{\sigma}}$ and $-{\boldsymbol{\sigma}}$, respectively. See Sec. \ref{['section:illustrative_derivation']} for more details.
• Figure 4: Excess fields: Each of the four vertically-stacked panels is a schematic plot of the microscopic charge density ${\rho(x)}$ of a different one dimensional material, with positive and negative charge(s) coloured red and blue, respectively. If, at each value of $x$, we calculate the average, ${\expval{\rho}_a(x)}$, of all charge within a distance ${a/2}$ of $x$, we find that it vanishes everywhere in the green-shaded 'bulk' of each material, but is finite in the white surface regions. At each surface, the integral of ${\expval{\rho}_a(x)}$ over all points that are not in the bulk, but are within a distance $a$ of it, is the surface's excess of charge, ${\boldsymbol{\sigma}}$. The symbols $+$, $-$, and $0$ next to each surface indicate whether ${\boldsymbol{\sigma}}$ is positive, negative, or zero, respectively. The macroscopic analogue, ${\boldsymbol{\varrho}}$, of $\rho$ is defined, to a finite precision ${{\varepsilon_{\varrho}}}$, as its mesoscale average $\bar{\rho}$. ${{\boldsymbol{\varrho}}}$ vanishes everywhere in the bulk, but not at interfaces, in general. Therefore, because spatial averaging conserves charge, the excess charge at an interface, ${\boldsymbol{\sigma}}$, is simply the integral of ${\boldsymbol{\varrho}}$ across it.
• Figure 5: A 2D macrostructure: An electrical microstructure in ${A\subset\mathbb{R}^2}$ is an areal charge density ${\sigma:A\to\mathbb{R}; (y,z)\mapsto \sigma(y,z)}$, which may be the excess of a volumetric charge density ${\rho(x,y,z)}$ on a surface or interface normal to the ${x}$-axis. An excess field (see Fig \ref{['fig:excess_fields']} and Sec. \ref{['section:excess_fields']}) is created at an interface whenever a higher dimensional microstructure is homogenized along an axis normal to the interface. The macrostructure of $A$ consists of continua that are punctuated with, and separated by, subspaces ${s_i}$ of dimensions zero or one, on which excess fields are defined. If $s_i$ has dimension one its excess field is a linear charge density ${\mathbfcal{L}_i}$; and if it has dimension zero its excess field is a point charge ${\mathbf{q}^{ (i)}}$.

Theorems & Definitions (3)

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