  The dynamics of the state-switching process of black holes in dRGT massive
gravity theory is presented using free energy landscape and stochastic Langevin
equations. The free energy landscape is constructed using the Gibbons-Hawking
path integral method. The black hole phases are characterized by taking its
horizon radius as the order parameter. The free energy landscape provides three
black hole phases: small, intermediate, and large. The small and large black
holes are thermodynamically stable whereas the intermediate one is unstable.
The Martin-Siggia-Rose-Janssen-de Dominicis (MSRJD) functional describes the
stochastic dynamics of black hole phase transition. The Hamiltonian flow lines
are obtained from the MSRJD functional and are used to analyze the stability
and the phase transition properties. The dominant kinetic path between
different phases is discussed for various configurations of the free energy
landscape. We discuss the effect of black hole charge and the graviton mass on
the critical behavior of black hole phase transition.
