Path integral and instantons for the dynamical process and phase transition rate of Reissner-Nordström-AdS black holes

We propose a new approach to study the phase transition dynamics of the Reissner-Nordstr\"om-AdS black holes on the underlying free energy landscape. By formulating a path integral framework, we quantify the kinetic paths representing the histories from the initial state to the end state, which provides us a visualized, yet quantified, picture about how the phase transition proceeds. Based on these paths, we derive the analytical formulas for the time evolution of the transition probability and provide a physical interpretation of the contribution to the probability from one pseudomolecule (antipseudomolecule), composed of instantons and anti-instantons, which is actually the phase transition rate from the small(large) to the large(small) black hole state. These numerical results show a good consistency with the underlying free energy landscape topography.