Path integral and instantons for the process and phase transition rate of the RNAdS black hole

Abstract

We propose a new approach to study the dynamical phase transition of RNAdS black holes on the underlying free energy landscape. By formulating a path integral framework, we can quantify the paths representing the history from the initial state to the end state of the phase transition. Then, the dynamics can be reflected completely. Based on these paths, we can calculate the kinetic rates and obtain the analytical expressions through the sums of the instantons. We find that the phase transition rate increases (decreases) with the temperate from the small (large) to large (small) black hole state transition. This is consistent with the change of the free energy landscape topography upon the temperature. Furthermore, the time evolution of the probability of starting from the small black hole initially and ending at the small (or large) black hole P (r s , t; r s , 0) (or P (r l , t; r s , 0)) is also given in this study.

Non-Markovian dynamics of black hole phase transition

“…This is the free energy landscape description of the RNAdS black holes. This description is universal in studying the black hole phase transition [16][17][18][19][20][21][22][23][24][25][26][27][28].…”

“…On the free energy landscape, we are allowed not only to analyze the thermodynamically favored state but also to characterize the dynamical process of the black hole transition from one state to another state. One can refer to references [16][17][18][19][20][21][22][23][24][25][26][27][28] for some related works.…”

Generalized free energy landscape of black hole phase transition

“…This constraint implies that the RNAdS black hole cannot have arbitrary small event horizon due to the presence of electric charge [18].…”

“…In this way, one can study the kinetics of black hole phase transition by using the Langevin equation for the dynamical trajectory evolution or the Fokker-Planck equation for the probabilistic evolution. The formalism has been applied to investigate the Markovian dynamics [8][9][10][11][12][13][14][15][16][17][18][19][20][21] and the non-Markovian dynamics [22,23] of the black hole phase transitions.…”

Kinetics of Hawking-Page phase transition with the non-Markovian effects

“…The third force represents the stochastic force that comes from the microscopic degrees of freedom of the effective heat bath. Under these assumptions, the Langevin equation or the equivalent Fokker-Planck equation are employed to calculate the kinetic time and reveal the dynamical properties of the black hole phase transitions [35][36][37][38][39][40][41][42][43][44][45][46][47]. In [34], the possibility of probing the black hole microstructure from the kinetic turnover of the small/large RNAdS black hole phase transition was discussed.…”