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.
Highlights
We explore the thermodynamics and the underlying kinetics of the van der Waals type phase transition of Reissner-Nordstrom 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
In this work, inspired by the analogy of RNAdS black holes and van der Waals liquid system, we will go a few steps further and gain deeper understanding on this by studying the thermodynamics and the underling kinetics of the phase transition between the small and the large RNAdS black holes based on the free energy landscape
Summary
We start with the metric of RNAdS black hole, which describes the spherically symmetric charged black hole solution to Einstein-Maxwell action with negative cosmological constant [61]. In the case of an asymptotically AdS black hole in four dimensions, one can relate the thermodynamic pressure to the cosmological constant as suggested in [8,9,10]. In figure 1, we have depicted the black hole temperature as a function of black hole radius r+ when P < Pc (left pannel) and P > Pc (right pannel). When P < Pc, the local minimal and local maximum values of black hole temperature are determined by equation. When substituting back to eq (2.6), one can get the local minimal and local maximum values of black hole temperature which are respectively given by. When Tmin < TH < Tmax, there exists three branches of black hole solution (i.e. small, intermediate, and large black hole). Similar to the van der Waals liquid-gas system, there is a first order phase transition from the small black hole to the large black hole
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