  We provide a comprehensive study on the non-Markovian dynamics of the black
hole phase transitions based on the underlying free energy landscape. By
assuming that the transition processes between different black hole states are
stochastic, the non-Markovian dynamics of the black hole phase transition is
governed by the generalized Langevin equation with the time-dependent friction
that represents the memory effect from the effective thermal bath when the
timescale of the system is comparable or shorter than the timescale of the
effective thermal bath. We consider the first passage problem associated with
the black hole phase transitions and derive the analytical expressions of the
mean first passage time in the weak, intermediate, and large friction regimes.
As the concrete examples, we study the effects of three types of time dependent
friction kernel (delta function friction, exponentially decayed friction, and
oscillatory friction) on the dynamics of Hawking-Page phase transition and the
small/large RNAdS black hole phase transition. we found that there is a kinetic
turnover point for each type of friction kernel when the friction strength
varies. For the exponentially decayed friction kernel, it is shown that the
non-Markovian effect slows down the phase transition dynamics in the weak
friction regime and speeds up the transition process in the strong friction
regime. For the oscillating decayed friction kernel, we found kinetic
resonances when the oscillating frequency of the effective thermal bath is
equal to the oscillating frequency of the black hole state in the initial
potential well on the free energy landscape.
