the Taylor expansion of energy with respect to $u_{i}$, which is the local dipole on site $i$. For a homogeneous system, where $k_{i}$ and $\lambda_{i}$ are constants, we can usually first consider the harmonic term and construct phonon modes, which are then used to describe the system, especially in low temperature when the system condense to particular phonon modes [@CondenedPhononMode]. In contrast, with the loss of periodicity in relaxors, this approach is no longer profitable. One can insist on using averaged atoms (e.g., replacing Zr and Ti atoms with their average in BZT) to retain the use of phonon modes. However, it is then necessary to consider defect-pinned intrinsic localized modes [@PinnedPhonon] and phonon localization [@PhononLocalization].

The inhomogeneity also has important consequences on ferroelectric phase transitions. In the typical ferroelectric material BaTiO$_{3}$, we may ascribe the temperature-driven phase transition to the condense of phonons to a particular phonon mode [@CondenedPhononMode]. At high temperature, many phonons modes are occupied (occupancy obeying the Bose-Einstein distribution); at low temperatures, due to mode softening, certain mode (often corresponding to the well-known in perovskites) has essentially zero energy, which dominates the system and induce phase transitions. Unlike BaTiO$_{3}$, there is no global phase transition due 