  Historically, the soft mode theory of ferroelectric phase transitions has
been developed for the high-temperature (paraelectric) phase, where the phonon
mode softens upon decreasing the temperature. In the low-temperature
ferroelectric phase, a similar phonon softening occurs, also leading to a
bosonic condensation of the frozen-in mode at the transition, but in this case
the phonon softening occurs upon increasing the temperature. Here we present a
soft mode theory of ferroelectric and displacive phase transitions by
describing what happens in the low-temperature phase in terms of phonon
softening and instability. A new derivation of the generalized
Lyddane-Sachs-Teller (LST) relation for materials with strong anharmonic phonon
damping is also presented which leads to the expression
$\varepsilon_{0}/\varepsilon_{\infty}=|\omega_{LO}|^{2}/|\omega_{TO}|^{2}$. The
theory provides a microscopic expression for $T_c$ as a function of physical
parameters, including the mode specific Gr\"uneisen parameter. The theory also
shows that $\omega_{TO} \sim (T_{c}-T)^{1/2}$, and again specifies the
prefactors in terms of Gr\"uneisen parameter and fundamental physical
constants. Using the generalized LST relation, the softening of the TO mode
leads to the divergence of $\epsilon_0$ and to a polarization catastrophe at
$T_c$. A quantitative microscopic form of the Curie-Weiss law is derived with
prefactors that depend on microscopic physical parameters.
