2N\sqrt{\frac{E_{\textrm{kin}}}{\pi}}\left(\frac{1}{k_{B}T}\right)^{3/2}\exp\left(-\frac{E_{\textrm{kin}}}{k_{B}T}\right),\label{eq:kinetic-energy}\end{aligned}$$ where $k_{B}$ is the Boltzmann constant, $N$ is the total number of dipoles, and $f\left(E_{\textrm{kin}}\right)dE_{\textrm{kin}}$ is the number of dipoles having a kinetic energy between $E_{\textrm{kin}}$ and $E_{\textrm{kin}}+dE_{\textrm{kin}}$. With this distribution function, we can calculate the number of dipoles with kinetic energy that exceeds the potential well $E_{b}$, which is given by $$\begin{aligned}
N_{1}\left(E_{b},T\right) & =\int_{E_{b}}^{\infty}dE_{\textrm{kin}}f\left(E_{\textrm{kin}}\right)\nonumber \\
= & N\sqrt{\frac{4}{\pi}}\sqrt{\frac{E_{b}}{k_{B}T}}\exp\left(-\frac{E_{b}}{k_{B}T}\right)+N\text{erfc}\left(\sqrt{\frac{E_{b}}{k_{B}T}}\right),\label{eq:particles-above-well}\end{aligned}$$ where erfc is the complementary error function. The number of dipoles confined to the potential well is then given by $$\begin{aligned}
N_{2}\left(E_{b},T\right)= & N-N_{1}\left(E_{b},T\right).\label{eq:particles-inside-well}\end{aligned}$$

The next step is to treat the two sets of dipoles ($N_{1}$ versus $N_{2}$) separately, assigning different susceptibility to them. The total susceptibility is then given by $$\begin{aligned}
\chi(T,\nu)= & \chi_{1}\left(T,\nu\right)P_{1}\left(E_{b},T\right)+\chi_{2}\left(T,\nu\right)P_{2}\left(E_{b},T\right),\label{eq:total-susceptibility}\end{aligned}$$ where $\chi_{1}\left(T,\nu\right)$ and $\chi_{2}\left(T,\nu\right)$ describes the dielectric responses of each dipole group, whose form will be specified later. We also define $$\begin{aligned}
P_{1}\left(E_{b},T\right)\equiv & \frac{1}{N}N_{1}\left(E_{b},T\right),\\
P_{2}\left(E_{b},T\right)\equiv & \frac{1}{N}N_{2}\left(E_{b},T\right),\end{aligned}$$ to normalize the dipoles to unit volume. Equation (\[eq:total-susceptibility\]) is the centerpiece of this work and will be demonstrated to be useful for investigation of various relaxors.

Results \[sec:Results\]
=======================

We now apply Eq. (\[eq:total-susceptibility\]) to fit various $\chi$ versus $T$ obtained experimentally or numerically. The relaxors shown here include both lead-basd (e.g., PMN) and lead-free 