Thermodynamic Phase Transition of AdS Black Holes in Massive Gravity on Free Energy Landscape

Black hole (BH) thermodynamics is a research field integrating gravitational theory, thermodynamics and quantum theory, which has been the focus of attention. The phase transition (PT) of BHs is an important part of BH thermodynamics. Through the study of BH PT, we will have a deeper understanding of the nature of BHs, and the gravity theory related to BHs will be developed. In the study of the influence of massive gravity parameters on PT dynamics, we study the thermodynamic PT process of AdS BHs in massive gravity in detail. We compare the cases with different massive gravity parameters. Based on the Gibbs free energy landscape, we study the kinetic properties of PT. We find that the two wells have the mallme depth at a certain temperature on the free energy landscape. Due to thermal fluctuations, the large black hole (LBH) state and small black hole (SBH) state can be transitioned to each other. By solving the Fokker-Planck equation, we study the probabilistic evolution of the PT of AdS BHs in massive gravity. The results show that the initial state of the BH evolves faster with the increase of the parameter \(\varvec{\lambda }\). In addition, for SBH state and LBH state, the mean first passage time increases with the increase of the parameter \(\varvec{\lambda }\). Considering that the mean first passage time of the PT of AdS BHs in massive gravity can be obtained, this work can play an important role in limiting the massive gravity parameter according to observations in the future.