10. Charge density ($\Rho$) and dipole moment density ($\mbp$)
https://doi.org/10.48550/arXiv.2403.13981
In this section I consider the mesoscale averages of charge and dipole moment densities. For simplicity I define the mesoscale average as the simple average introduced and discussed in detail in Sec. 8.
10.1 Charge density
If we ignore the microscale variability of the mesoscale average, $\bar{\rho}$, of $\rho$, and the consequent uncertainty, $\precRho$, in the value of the macroscopic charge density, its definition is simply \begin{align*}
\Rho(\bx) \equiv \bar{\rho}(x) = \frac{1}{\intmax}\int_{-\intmax/2}^{\intmax/2}\rho(x+u)\,\dd{u}
\tag{54}
\end{align*}
where ${\intmax \gg \amax}$.
In the bulk of a crystal, $\intmax$ can be chosen to be an integer multiple of the periodicity, $a$, where ${\rho(x+a)=\rho(x),\;\;\forall\,x\in\bulk}$. It is then easy to show that
\begin{align*}
\Rho(\mxb)= \bar{\rho}(x_b) = \frac{1}{a} \int_{-a/2}^{a/2}\rho(x_b + u)\,\dd{u}.
\end{align*}
This vanishes if the crystal is charge-neutral, as expected of $\Rho$. In amorphous materials, if $\rho$ fluctuates microscopically about zero, it is always possible to find microscopic displacements, ${\eta_1\sim a}$ and ${\eta_2\sim a}$, such that \begin{align*}
\int_{-\intmax/2+\eta_1}^{\intmax/2+\eta_2}\rho(x_b+u)\,\dd{u}=0.
\end{align*}
By expressing the integral in Eq. (54) as \begin{align*}
\int_{-\intmax/2}^{\intmax/2}=\int_{-\intmax/2}^{-\intmax/2+\eta_1}+ \int_{-\intmax/2+\eta_1}^{\intmax/2+\eta_2} -\int_{\intmax/2}^{\intmax/2-\eta_2}
\end{align*}
it is straightforward to show that \begin{align*}
\bar{\rho}(x_b)=0 + \order{\amax/\prectheo}\approx 0.
\end{align*}
Therefore, to within the finite precision $\precRho$ with which $\Rho$ can be defined, $\Rho$ vanishes in the bulk of any material that is stable when it is electromagnetically isolated, and whose surfaces are locally charge neutral.
10.2 Dipole moment density
$\pp$ has the dimensions of a dipole moment per unit volume, area, and length in three, two, and one dimensions, respectively. Therefore, to define $\pp$ within the bulk of a one dimensional material, it seems natural to start from the quantity \begin{align*}
\mpp(x,\epsilon) \equiv \frac{1}{\epsilon}
\int_{-\epsilon/2}^{\epsilon/2} \rho(x+u)\, u \, \dd{u}, \tag{55}
\end{align*}
which is the dipole moment per unit length of ${\interval(x,\epsilon)\subset\bulk}$ with respect to an origin at $x$. $\mpp(x,\epsilon)$ is strongly dependent on both $x$ and $\epsilon$ and so it is difficult to attach physical meaning to it. However, it is clearly a microscopically-varying quantity and its mesoscale average is \begin{align*}
\bar{\mpp}(x_b) & =
\frac{1}{\intmax}\int_{x_b-\intmax/2}^{x_b+\intmax/2}
\left(\frac{1}{\epsilon}\int_{-\epsilon/2}^{\epsilon/2}
\rho(x+u)\,u \,\dd{u}\right) \;\dd{x} \\
& =
\frac{1}{\epsilon}\int_{-\epsilon/2}^{\epsilon/2}
u \left(\frac{1}{\intmax} \int_{x_b-\intmax/2}^{x_b+\intmax/2} \rho(x+u)\, \dd{x}\right)\, \dd{u} \\
& =
\frac{1}{\epsilon}\int_{-\epsilon/2}^{\epsilon/2} u
\,\bar{\rho}(x_b+u) \dd{u} \\
& =
\frac{\bar{\rho}(x_b)}{\epsilon}
\int_{-\epsilon/2}^{\epsilon/2} u \;\dd{u}
+\order{\amax/\prectheo}
\approx 0,
\tag{56}
\end{align*}
where, by using ${\bar{\rho}(x_b+u)=\bar{\rho}(x_b) + \order{\amax/\prectheo}}$, I am assuming that ${\bar{\rho}}$ fluctuates microscopically but does not change systematically on length scale $\epsilon$. Therefore, the mesoscale average $\bar{\mpp}$ of $\mpp$ is negligible when ${\amax/\prectheo}$ is sufficiently small, regardless of the value of ${\bar{\rho}}$.
This result generalises to three dimensions, where it can be shown that each Cartesian component of the mesoscale average of the dipole moment per unit volume of a region of arbitrary shape scales like ${\amax/\prectheo}$. This is a generalisation to non-crystalline materials of the well known result that, in a crystal, the average over all choices of unit cell of the dipole moment per unit cell is zero [Resta and Vanderbilt, 2007].
These results suggest that ${\mbp\equiv\bar{\mpp}}$ is not a useful macroscopic quantity with which to characterise the bulk of a material because it does not distinguish between different mesoscopically-uniform materials, or even between a material and empty space. We can only identify the macroscopic polarization $\pp$ as $\mbp$ if we are willing to accept that ${\pp=0}$ in every mesoscopically-uniform material, regardless of its microstructure.
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