increases with temperature, while the number of dipoles inside ($P_{2}$) continuously becomes smaller. Figure \[fig:components\] (b) shows that the Lagevin function is normalized at $T=0$, and decreases with temperature. Such a feature describes the ability of the free dipoles to respond to an external *dc* electric field. Moreover, Fig. \[fig:components\] (b) also shows the product of the Langevin function and $P_{1}$, which already exhibits some resemblance to BZT’s susceptibility [\[]{}Fig. (\[fig:Fitting-susceptibility-of\])[\]]{}.

![Fitting the susceptibility of Ba(Zr$_{0.5}$,Ti$_{0.5}$)O$_{3}$ at $f=1,10,100,100$ GHz, obtained from molecular dynamics simulation using Eq. (\[eq:BZT-frequency\]).\[fig:Fitting-BZT-frequency\]](Fig2){width="8cm"}

Having examined the static susceptibility, we now move to the frequency-dependent dielectric response, which is often taken as a characteristic property of relaxors [@Kleeman2014; @Sommer]. We propose another equation to fit the susceptibility versus temperature:

$$\begin{aligned}
\chi\left(T\right)= & \frac{\chi_{1}}{1+b\exp\left(-\theta/T\right)}P\left(E_{b},T\right)+\chi_{2}\left[1-P\left(E_{b},T\right)\right],\label{eq:BZT-frequency}\end{aligned}$$

where $\chi_{1}$ , $\chi_{2}$, $b$ and $\theta$ are constants at a given frequency (but may change when the frequency changes). The dielectric contribution from dipoles with kinetic energy higher than the potential well is given by $$\begin{aligned}
w_{1}\left(T\right)= & \frac{1}{1+b\exp\left(-\theta/T\right)}.\label{eq:h1T}\end{aligned}$$ which, similar to the Langevin function, monotonically decreases with temperature $T$. The choice of this function reflects two considerations: (i) at very low temperature ($T$ close to 0), such dipoles shall follow the probing 