  The soft modes associated with continuous-order phase transitions are
associated with strong anharmonicity. This leads to the overdamped limit where
the phonon quasi-particle picture can breakdown. However, this limit is
commonly restricted to a narrow temperature range, making it difficult to
observe its signature feature, namely the breakdown of the inverse relationship
between the relaxation time and damping. Here we present a physically intuitive
picture based on the relaxation times of the mode coordinate and its conjugate
momentum, which at the instability approach infinity and the inverse damping
factor, respectively. We demonstrate this behavior for the cubic-to-tetragonal
phase transition of the inorganic halide perovskite CsPbBr$_3$ via molecular
dynamics, and show that the overdamped region extends almost 200 K above the
transition temperature. Further, we investigate how the dynamics of these soft
phonon modes change when crossing the phase transition.
