model to both lead-free and lead-based relaxors. In Sec. \[subsec:Discussion\], we discuss the implication and limitation of this approach. Finally, in Sec. \[sec:Conclusion\], we present a brief conclusion.

Statistical modeling\[sec:Statistical-modeling\]
================================================

The statistical model starts by considering a critical difference between ferroelectric relaxors and normal ferroelectrics. One crucial observation is that all relaxor ferroelectrics discovered so far are inevitably disordered and inhomogeneous systems. For instance, in BZT Zr and Ti ions are distributed randomly, so are the Mg and Nb ions in PMN, when the samples are treated macroscopically. In addition, PMN possesses the electric field arising from heterovalent Mg and Nb ions, which affects dipole distribution. It is important to further note that well known relaxors can become non-relaxor if their ions are perfectly ordered [@Setter1980; @Bokov1984; @WangJAD].

Individual dipoles
------------------

The randomness of ions and the ensuing lack of long-range correlation has the important consequence that phonon modes may not be the best description to understand relaxor. This fact is evidenced by the effective Hamiltonian that describes the BZT relaxor [@Akbarzadeh2012; @Sherrington2013a; @Sherrington2014] $$\begin{aligned}
E= & \sum_{i}\left(\kappa_{i}\left|u_{i}\right|^{2}+\lambda_{i}\left|u_{i}\right|^{4}\right)+\dots,\label{eq:effHBZT}\end{aligned}$$ where $i$ labels the sites occupied randomly by Zr or Ti, and $\kappa_{i}$ ($\lambda_{i}$) are the second (fourth) order coefficients in 