  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 state switching
process. By solving the Hamiltonian equations of motion, we obtain the kinetic
path in the phase space using an example of the RNAdS 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 SAdS black hole to the thermal AdS 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.
