to the existence of PNRs and/or random electric fields, which eliminates the mode softening phenomenon and renders a global dipole pattern difficult to achieve [@Akbarzadeh2012; @PMNeffHami2015]. In addition, relaxors exhibit strange phonon behavior, such as the “waterfall” effect [@Gehring2000; @Hlinka2003; @PMNWaterFall] and the localization[@PhononLocalization], further showing their difference from normal ferroelectrics. In this work, we focus on individual dipoles and statistically model their dielectric response. This change of view point implies that phonon modes are less important in our analysis. We will show in the following that such change leads to fruitful results, and better understanding of relaxors.

Statistics of individual dipoles
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Individual dipoles can be categorized into different groups based on their dynamics, and each group shall have different contribution to susceptibility. We proceed simplify the interaction between dipoles, assuming that the interaction effectively introduce a potential well of *average* depth, $E_{b}$. We may relate $E_{b}$ to the size of PNRs arising from the clustering of same-type ions [@Akbarzadeh2012] and/or random electric field caused by heterovalent ions [@PMNeffHami2015; @Kleemann2014; @PMNWaterFall].

Since the kinetic energy of individual dipoles obeys the Maxwell-Boltzmann distribution, at temperature $T$, the number of dipoles with kinetic energy $E_{\textrm{kin}}$ is given by $$\begin{aligned}
f\left(E_{\textrm{kin}}\right)= & 