Thermal dynamic phase transition of Reissner-Nordström Anti-de Sitter black holes on free energy landscape

Abstract

We explore the thermodynamics and the underlying kinetics of the van der Waals type phase transition of Reissner-Nordström anti-de Sitter (RNAdS) black holes based on the free energy landscape. We show that the thermodynamic stabilities of the three branches of the RNAdS black holes are determined by the underlying free energy landscape topography. We suggest that the large (small) RNAdS black hole can have the probability to switch to the small (large) black hole due to the thermal fluctuation. Such a state switching process under the thermal fluctuation is taken as a stochastic process and the associated kinetics can be described by the probabilistic Fokker-Planck equation. We obtained the time dependent solutions for the probabilistic evolution by numerically solving Fokker-Planck equation with the reflecting boundary conditions. We also investigated the first passage process which describes how fast a system undergoes a stochastic process for the first time. The distributions of the first passage time switching from small (large) to large (small) black hole and the corresponding mean first passage time as well as its fluctuations at different temperatures are studied in detail. We conclude that the mean first passage time and its fluctuations are related to the free energy landscape topography through barrier heights and temperatures.

Observing dynamic oscillatory behavior of triple points among black hole thermodynamic phase transitions

“…We report that F P (t) is governed not only by the barrier height, as discussed previously [19,20], but also by the barrier width. An inspection of Figure 3 indicates that the difference in barrier heights ∆G L = G m1 − G m2 ≃ 0.001, whereas the difference in barrier widths is ∆r m2 −∆r m1 ≃ 0.02, 20 times larger than ∆G L .…”

“…where H and S are the respective enthalpy and entropy. This generalized off-shell free energy describes transient black hole states with r h as the order parameter, describing the underlying microscopic degrees of freedom [20]. We illustrate G L at the triple point in Figure 3 others are off-shell transient states [22].…”

“…This dynamic phase transition process at the triple point is different from that of a VdW type phase transition [20][21][22]. We now examine the detailed evolution of each coexistent black hole phase by plotting in Figure 5 the behavior of ρ(r k , t), when the initial Gaussian wave packet peaks at these states.…”

“…We seek here more clues about these oscillatory phenomena from the distribution of first passage times of the phase transition. This is defined as the rate at which a given initial state first reaches an unstable black hole phase, represented by the peak of the free energy [20]. From Figure 3, we have three cases: small to intermediate (case I), intermediate to small and large (case II), and large to intermediate (case III).…”

“…The phase transitions including the small/large black hole phase transition have been understood using the quasinormal modes of black holes [8][9][10][11][12][13][14][15][16][17][18]. Recent studies of the dynamics of phase transitions have been conducted [19][20][21] by solving a special case of the Fokker-Planck equation known as the Smoluchowski equation (SE). A dynamic transition indeed occurs between the unstable and stable black hole phases, and further investigation confirmed that even between stable small and large black hole phases, the dynamic process could occur [22].…”