Phase space path integral approach to the kinetics of black hole phase transition

We employ the approach of path integral in the ``phase space'' to study the kinetics of state switching associated with black hole phase transitions. Under the assumption that the state switching process of the black hole is described by the stochastic Langevin equation based on the free energy landscape, we derived the Martin--Siggia--Rose--Janssen--de Dominicis (MSRJD) functional and obtained the path integral expression of the transition probability. The MSRJD functional inherently represents the path integral in the ``phase space,'' allowing us to extract the effective Hamiltonian for the dynamics of the state switching process. By solving the Hamiltonian equations of motion, we obtain the kinetic path in the ``phase space'' using an example of the Reissner-Nordstrom anti-de Sitter black hole. Furthermore, the dominant kinetic path within the configuration space is calculated. We also discuss the kinetic rate by using the functional formalism. Finally, we examine two further examples: Hawking-Page phase transition and Gauss-Bonnet black hole phase transition at the triple point. Our analysis demonstrates that, concerning the Hawking-Page phase transition, while a dominant kinetic path in the ``phase space'' from the large Schwartzschild anti-de Sitter black hole to the thermal anti--de Sitter space is present, there is no kinetic path for the inverse process. For the Gauss-Bonnet black hole phase transition at the triple point, the state switching processes between the small, the intermediate and the large Gauss-Bonnet black holes constitute a ``chemical reaction cycle.''