Non-Markovian dynamics of black hole phase transition

Without loss of generality, we consider the Schwarzschild-AdS (SAdS) black hole in four dimensions. The metric is given by Hawking:1982dh

$\displaystyle ds^{2}=-\left(1-\frac{2M}{r}+\frac{r^{2}}{L^{2}}\right)dt^{2}+\left(1-\frac{2M}{r}+\frac{r^{2}}{L^{2}}\right)^{-1}dr^{2}+r^{2}d\Omega_{2}^{2}\;,$

(1)

where $M$ and $L$ are the black hole mass and the AdS curvature radius. When $M=0$, the metric (1) gives the metric of pure AdS space in four dimensions. The mass, the Hawking temperature, and the Bekenstein-Hawking entropy of the SAdS black hole are respectively given by

	$\displaystyle M$	$\displaystyle=$	$\displaystyle\frac{r_{+}}{2}\left(1+\frac{r_{+}^{2}}{L^{2}}\right)\;,$		(2)
	$\displaystyle T_{H}$	$\displaystyle=$	$\displaystyle\frac{1}{4\pi r_{+}}\left(1+\frac{3r_{+}^{2}}{L^{2}}\right)\;,$		(3)
	$\displaystyle S$	$\displaystyle=$	$\displaystyle\pi r_{+}^{2}\;,$		(4)

where $r_{+}$ is the black hole radius. The black hole radius is determined by the equation $1-\frac{2M}{r}+\frac{r^{2}}{L^{2}}=0$ and is given by the largest root of this equation.

Besides the thermal AdS space and the small/large SAdS black holes, which are the solutions to the Einstein equations, we also assume that there are black holes with arbitrary horizon radii generated by the thermal fluctuations. These black holes are the fluctuating black holes. Following the discussion in Li:2020khm , we consider the canonical ensemble at the specific temperature $T$, which is composed of a series of black hole spacetimes with an arbitrary horizon radius. We call the black hole spacetime in the ensemble as the black hole state. The black hole state in the ensemble is then characterized by the black hole radius, which is considered as the order parameter. According to the ensemble theory, every black hole state in the ensemble has the specific probability, which is determined by the generalized free energy function. Our aim is to construct the relationship between the generalized free energy and the order parameter.

Our starting point is the on-shell Gibbs free energy of the SAdS black hole. According to the Euclidean gravitational path integral Gibbons:1976ue , the on-shell Gibbs free energy for the black hole solution to the Einstein equation can be calculated directly from the Euclidean Einstein-Hilbert action, or from the thermodynamic relationship $G=M-T_{H}S$. In order to quantify the free energy landscape, we need to specify every spacetime state in the ensemble a Gibbs free energy. We generalize the on-shell Gibbs free energy to the off-shell Gibbs free energy by replacing the Hawking temperature $T_{H}$ with the ensemble temperature $T$ in the thermodynamic relationship, which is explicitly given as follows York:1986it ; Spallucci:2013osa ; Andre:2020czm ; Andre:2021ctu

$\displaystyle G=M-TS=\frac{r_{+}}{2}\left(1+\frac{r_{+}^{2}}{L^{2}}\right)-\pi Tr_{+}^{2}\;.$

(5)

In the above equation, the black hole radius $r_{+}$ is treated as the order parameter, which is emergent from the underlying microscopic degrees of freedom of the black hole. The order parameter $r_{+}$ in the above equation changes continuously. The generalized free energy is then the continuous function of the order parameter and ensemble temperature.

In Figure 1, the free energy landscapes for the Hawking-Page phase transition are plotted for different ensemble temperatures. The shape of the free energy landscape for the Hawking-Page phase transition is only modulated by the ensemble temperature $T$. In these plots, the origin $r_{+}=0$ represents the pure radiation phase or thermal AdS space. When $T<T_{min}=\frac{\sqrt{3}}{2\pi L}$, the free energy is the monotonic increasing function of the order parameter and the origin $r_{+}=0$ is the global minimum on the free energy landscape. The system has only one stable phase, i.e. the thermal AdS space. At $T=T_{min}$, Gibbs free energy landscape exhibits an inflection point at $r_{+}=L/\sqrt{3}$. From this temperature on, the two black hole phases emerge (large and small black holes) with radii given by

$\displaystyle r_{l,s}=\frac{T}{2\pi T_{min}^{2}}\left(1\pm\sqrt{1-\frac{T_{min}^{2}}{T^{2}}}\right)\;.$

(6)

The small/large SAdS black hole phase is represented by the maximum/minimum point on the free energy landscape. The small black hole phase is always unstable while the large black hole phase can be stable or unstable depending on the temperature.

The stability of the large SAdS black hole can be analyzed in terms of the fact that the global or the local minimum on the free energy landscape corresponds to the global or the local stable phase. From the plots of the free energy landscapes, we can easily observe that below the Hawking-Page critical temperature $T=T_{HP}=\frac{1}{\pi L}$, the thermal AdS phase is thermodynamically stable, and above the critical temperature, the large black hole phase is thermodynamically stable. At the critical temperature, both the thermal AdS space phase and the large black hole phase are stable with equal free energy basin depth. Since there is a discontinuous change in the order parameter from the thermal AdS phase $r_{+}=0$ to the large black hole phase $r_{+}=r_{l}$ at the Hawking-Page critical temperature, the associated derivative of the free energy function will be divergent, which is a signature of the first order phase transition. It should be noted that besides the thermal AdS space and the small and large SAdS black holes represented by the maximum or the minimum points on the free energy landscape, there are fluctuating black holes on the free energy landscape. The fluctuating black holes are the intermediate states of the phase transition.

In the present subsection, we consider another example of the black hole phase transition based on the free energy landscape, i.e. the small/large RNAdS black hole phase transition Kubiznak:2012wp . The metric of RNAdS black hole in four dimensions is given by

$\displaystyle ds^{2}=-\left(1-\frac{2M}{r}+\frac{Q^{2}}{L^{2}}+\frac{r^{2}}{L^{2}}\right)dt^{2}+\left(1-\frac{2M}{r}+\frac{Q^{2}}{L^{2}}+\frac{r^{2}}{L^{2}}\right)^{-1}dr^{2}+r^{2}d\Omega_{2}^{2}\;,$

(7)

where $M$ is the mass, $Q$ is the electric charge, and $L$ is the AdS curvature radius.

The mass, the Hawking temperature, and the Bekenstein-Hawking entropy of RNAdS black hole are given by

	$\displaystyle M$	$\displaystyle=$	$\displaystyle\frac{r_{+}}{2}\left(1+\frac{r_{+}^{2}}{L^{2}}+\frac{Q^{2}}{L^{2}}\right)\;,$		(8)
	$\displaystyle T_{H}$	$\displaystyle=$	$\displaystyle\frac{1}{4\pi r_{+}}\left(1+\frac{3r_{+}^{2}}{L^{2}}-\frac{Q^{2}}{L^{2}}\right)\;,$		(9)
	$\displaystyle S$	$\displaystyle=$	$\displaystyle\pi r_{+}^{2}\;,$		(10)

where $r_{+}$ is the black hole radius.

Analogous to the case of Hawking-Page phase transition, we consider the canonical ensemble at the specific temperature $T$, which is constructed by including the three branches of the RNAdS black holes (the small, the large, and the intermediate black holes) as well as the fluctuating black holes Li:2020nsy ; Li:2021vdp . The black holes in the ensemble are also distinguished or described by the continuous order parameter, the horizon radius of the black hole. The three branches of the RNAdS black hole solutions are the solutions to the Einstein field equations, while the fluctuating black holes, which are generated by the thermal noise, are not necessarily the solutions to the Einstein field equations.

The on-shell Gibbs free energy of the three branches of the RNAdS black hole calculated from the Einstein-Hilbert action can be properly rewritten as the thermodynamic relationship of $G=M-T_{H}S$. We define the generalized off-shell Gibbs free energy of the fluctuating black hole as York:1986it ; Spallucci:2013osa ; Andre:2020czm ; Andre:2021ctu

$\displaystyle G=M-TS=\frac{r_{+}}{2}\left(1+\frac{8}{3}\pi Pr_{+}^{2}+\frac{Q^{2}}{r_{+}^{2}}\right)-\pi Tr_{+}^{2}\;,$

(11)

where $T$ is the ensemble temperature, $r_{+}$ is the radius of the fluctuating black hole or the order parameter, $P=-\frac{\Lambda}{8\pi}=\frac{3}{8\pi}\frac{1}{L^{2}}$ is an effective thermodynamic pressure with $\Lambda$ being the cosmological constant (negative value corresponding to AdS universe) and $L$ being the AdS curvature radius, and $Q$ is the charge of the black hole. Note that contrary to the case of Hawking-Page phase transition, we do not set the AdS radius $L$ to unity but treat it as the thermodynamic pressure $P$, which can be adjusted in the following discussion.

We quantify the free energy landscape of the RNAdS black hole by plotting the generalized Gibbs free energy as a function of black hole radius $r_{+}$ in Fig.2. It is shown that the shape of the free energy function varies along with the ensemble temperature. The detailed analysis of the free energy landscape topography can give rise to the phase diagram of the RNAdS black hole, which gives us all the information about the phase transition.

It can be observed that there exists two temperatures $T_{min}$ and $T_{max}$. When $T<T_{min}$ or $T>T_{max}$, the free energy landscape only has a single well. In this case, there is the small black hole phase or the large black hole phase respectively. When $T_{min}<T<T_{max}$, the free energy landscape exhibits the shape of the double well. The small and the large black holes are locally or globally stable states, while the fluctuating black holes and the intermediate black hole are unstable states. At the critical temperature $T=T_{PT}$, the small black hole phase and the large black hole phase have the equal free energy basin depth. This is the signature of the small/large black hole phase transition. On the free energy landscape, the unstable black holes are treated as the transient states, which are the bridges in the phase transition process. The probability of generating a fluctuating black hole is then determined by the generalized Gibbs free energy via the Boltzmann law $p(r_{+})\sim e^{-G(r_{+})/k_{B}T}$. In this sense, the generalized Gibbs free energy can be taken as the effective potential when studying the dynamics of the 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 Wei:2020rcd ; Li:2020spm ; Wei:2021bwy ; Cai:2021sag ; Lan:2021crt ; Li:2021zep ; Yang:2021nwd ; Mo:2021jff ; Kumara:2021hlt ; Li:2021tpu ; Liu:2021lmr ; Xu:2021usl ; Du:2021cxs .

In the previous Markovian models of the black hole phase transition based on the free energy landscape, it was assumed that the dynamics is govern by the Langevin equation Li:2021vdp . The Langevin equation that describes the evolution along the order parameter is considered to be the effective equation emergent from the microscopic degrees of freedom of the black hole. The basic idea comes from the Mori-Zwanzig formalism Mori:1965 ; Zwanzig:1960jcp .

Macroscopic system with a large number of microscopic degrees of freedom such as black hole can be well described by a small number of relevant variables. By applying the Mori-Zwanzig projection, one can find the macroscopic equations that only depend on the relevant variables from the microscopic equations of motion of a system. One can refer to Vrugt:2021sfu for the recent application of the Mori-Zwanzig formalism in the averaging problem in cosmology. The irrelevant microscopic degrees of freedom of black hole can be taken to be the effective thermal bath. In the generalized Langevin equation, it is assumed that the dynamics of the black hole phase transition is determined by the three types of forces. The first is the thermodynamic driving force determined by the generalized free energy function. The second force is the effective friction along the order parameter, which is interpreted as the interaction or the dissipation of the microscopic degrees of freedom from the effective heat bath acting on the black hole collective order parameter. The third force represents the stochastic force that comes from the microscopic degrees of freedom of the effective heat bath on the macroscopic order parameter.

In studying the non-Markovian model of black hole phase transition, the correlation time of the thermal bath is assumed to be comparable or even longer than the oscillating time of the black hole state at the potential well on the free energy landscape. The non-Markovian effect can be described by the time dependent friction kernel. Therefore, the non-Markovian dynamics can be described by the generalized Langevin equation complemented by the free energy potential as Mori:1965 ; Kubo:1966

	$\displaystyle\frac{dr}{dt}=v\;,$
	$\displaystyle\frac{dv}{dt}=-\frac{dG(r)}{dr}-\int_{0}^{t}\zeta(\tau)v(t-\tau)d\tau+\eta(t)\;,$		(12)

where the centered color noise $\eta(t)$ with zero mean and the dissipation kernel $\zeta(t)$ are related by the second fluctuation-dissipation theorem

$\displaystyle\zeta(|t|)=\frac{1}{k_{B}T}\langle\eta(0)\eta(t)\rangle\;.$

(13)

In Eq.(3.1), we have abbreviated the order parameter $r_{+}$ to $r$ for simplicity. As noted, the Langevin equation with the Markovian dynamics in Li:2021vdp can be reduced from Eq.(3.1) by taking the friction kernel instantaneous, i.e. $\zeta(t)=\zeta\delta(t)$. The non-Markovian effects is reflected by the time dependent friction kernel $\zeta(t)$. The generalized Langevin equation can be viewed as the effective equation that describes the evolution of the black hole order parameter at the macroscopic emergent level. It is an integro-differential equation.

Now, we will study the dynamics of black hole phase transition by treating the process as the first passage problem. First passage time is a very important quantity in characterizing the speed of the dynamics, which can be defined as the time required for one stable black hole state crossing the intermediate transient state to reach another stable state for the first time. On the free energy landscape, this problem can be regarded as the black hole state in the initial potential well crossing the barrier in the middle to reach the potential well on the other side. However, because of the stochastic nature of the transition process described by the Langevin equation, the first passage time is a stochastic quantity. The mean first passage time (MFPT) is then defined as the average timescale for a stochastic event to first occur. The MFPT is the characteristic time that describes the dynamics of the phase transition process.

In this section, we will derive the analytical expressions of the MFPT for the large friction case. Note that in deriving the analytical expressions of the MFPTs, we have assumed that the free energy landscape has the shape of double well as in the case of the RNAdS black holes. We mainly concern the transition rate or the MFPT of the dynamical transition process from the state in the left well to the state in the right well. When considering the Hawking-Page phase transition, the final results can be easily obtained by replacing the corresponding quantities in the RNAdS case with the corresponding ones in the Hawking-Page phase transition case.

In the large friction limit, the corresponding non-Markovian Fokker-Planck equation is in the form Sokolov:2002 ; Assis:2006

$\displaystyle\frac{\partial}{\partial t}\rho_{(}r,t)=\int D(t-\tau)\mathcal{L}\rho(r,t)d\tau\;,$

(14)

where $D(t)$ is the diffusion kernel and $\mathcal{L}$ is the linear operator acting on the order parameter $r$. The operator $\mathcal{L}$ has the same form as the Markovian model

$\displaystyle\mathcal{L}=\frac{\partial}{\partial r}e^{-\beta G(r)}\frac{\partial}{\partial r}e^{\beta G(r)}\;,$

(15)

where $\beta=1/k_{B}T$ is the inverse temperature. It should be noted that the diffusion kernel and the friction kernel have the relation as

$\displaystyle\int_{0}^{t}D(\tau)\zeta(t-\tau)d\tau=k_{B}T\delta(t)\;.$

(16)

The non-Markovian Fokker-Planck equation describes the time evolution of the probability distribution $\rho(r,t)$ in the ensemble. It is an integro-differential equation, which is obviously harder to deal with than the Fokker-Planck equation for the Markovian model. The non-Markovian Fokker-Planck equation (14) with a $\delta$-functional diffusion kernel $D(t)=D\delta(t)$ where $D=\frac{k_{B}T}{\zeta}$ corresponds to a usual Fokker-Planck equation describing the Markovian processes.

To calculate the MFPT, our starting point is the non-Markovian Fokker-Planck equation (14). However, as mentioned, it is difficult to deal with by using both the analytical and numerical methods. Firstly, we employ the Laplace transformation of the non-Markovian Fokker-Planck equation, which can be written in the frequency space as Bryngelson:1989

$\displaystyle s\rho(r,s)-n_{i}(r)=\hat{D}(s)\frac{\partial}{\partial r}e^{-\beta G(r)}\frac{\partial}{\partial r}e^{\beta G(r)}\rho(r,s)\;.$

(17)

This equation is similar to the Fokker-Planck equation in the frequency space that have been considered in Li:2020khm except the diffusion coefficient $D$ is replaced by the frequency dependent diffusion kernel $\hat{D}(s)$. Therefore, the frequency dependent diffusion kernel $\hat{D}(s)$ includes all the non-Markovian effects on the dynamics. From the relation between the diffusion kernel and the friction kernel, i.e., Eq.(16), the Laplace transformations give the relation in the frequency space as

$\displaystyle\hat{D}(s)\hat{\zeta}(s)=k_{B}T\;,$

(18)

which is analogous to the fluctuation-dissipation relation for the Markovian dynamics, while this relation is valid in the frequency space.

Defining $\Sigma(t)=\int_{0}^{r_{l}}\rho(r,t)dr$ as the probability that the state has not made a first passage by time $t$. Note that the first passage time is a random variable. The distribution of the first passage time is then given by $F_{p}(t)=-\frac{d\Sigma(t)}{dt}$. The MFPT defined as the average of the first passage time is then given by

$\displaystyle\hat{t}=\int_{0}^{+\infty}tF_{p}(t)dt=\int_{0}^{r_{l}}\rho(r,s=0)dr\;.$

(19)

Following the same procedure as done in Li:2020khm , one can derive the relation from the Fokker-Planck equation in frequency space

$\displaystyle\rho(r,s=0)=\frac{1}{\hat{D}(0)}\int_{r}^{r_{l}}dr^{\prime}\int_{0}^{r^{\prime}}dr^{\prime\prime}n_{i}(r^{\prime\prime})e^{\beta\left(G(r^{\prime})-G(r)\right)}\;.$

(20)

In deriving this equation, we have used the reflecting boundary condition at $r=0$ and the absorbing boundary condition at $r=r_{l}$. Finally, by substituting this equation into Eq.(19) and exchanging the order of integration, one can obtain the analytical integration expression for the MFPT of the transition process as Bryngelson:1989

$\displaystyle\bar{t}=\frac{1}{\hat{D}(0)}\int_{r_{i}}^{r_{l}}dr\int_{0}^{r}dr^{\prime}e^{\beta\left(G(r)-G(r^{\prime})\right)}\;,$

(21)

where $\hat{D}(0)$ is the zero frequency value of the frequency dependent diffusion kernel. In deriving this equation, the initial distribution $n_{i}(r)$ is selected to be $\delta(r-r_{i})$, where $r_{i}$ is taken to be $r_{i}=r_{s}$ representing the small RNAdS black hole for the small/large black hole phase transition. Then, Eq.(21) can be interpreted as the MFPT of the transition process from the small to the large RNAdS black hole phase. The MFPT for the inverse process can also be derived analogously Li:2020khm .

For the Hawking-Page phase transition, we only consider the transition process from the large SAdS black hole state in the right well to the thermal AdS state at $r=0$. Therefore, the reflecting boundary condition is imposed at $r=+\infty$ and the absorbing boundary condition is imposed at $r=0$. The expression of MFPT for this process is similar to the expression of MFPT for RNAdS case from the large black hole phase to the small black hole phase Li:2020khm .

There is an approximate formula for the MFPT in the large friction regime. Note that the Gibbs free energy near the minimum and the maximum can be approximated by the quadratic expansion as

	$\displaystyle G(r)$	$\displaystyle=$	$\displaystyle G(r_{s})+\frac{1}{2}\omega_{s}(r-r_{s})^{2}+\cdots$
	$\displaystyle\;,G(r)$	$\displaystyle=$	$\displaystyle G(r_{m})+\frac{1}{2}\omega_{m}(r-r_{m})^{2}+\cdots\;.$		(22)

By performing the Gaussian integral, the approximate expression of the MFPT in the high barrier limit $(\beta W\gg 1)$ can be obtained as

$\displaystyle\bar{t}=\frac{2\pi\hat{\zeta}(0)}{\omega_{s}\omega_{m}}e^{\beta W}\;.$

(23)

This formula of the MFPT is only valid for the case of large friction and high barrier. Note that this expression is valid for the RNAdS black hole phase transition. For the Hawking-Page phase transition, $\omega_{s}$ and $\omega_{m}$ should be replaced by $\omega_{l}$ and $\omega_{s}$ for the free energy landscape of SAdS black holes, and $W$ is the barrier height between the unsatble small SAdS black hole state and the large SAdS black hole state on the free energy landscape.

In the intermediate friction regime, the corresponding generalized Fokker-Planck equations for the Brownian oscillators were derived in Adelman:1976 . Note that the potential function used to derive the generalized Fokker-Planck equations is quadratic. The starting point is the probability distribution $\rho(r,v,t|r_{0},v_{0})$ with the initial parameters $r_{0}$ and $v_{0}$. Since the random force $\eta(t)$ is assumed to be Gaussian, the probability distribution $\rho(r,v,t|r_{0},v_{0})$ for the Brownian oscillators is shown to be also Gaussian, which is given by Adelman:1976

$\displaystyle\rho(r,v,t|r_{0},v_{0})=\frac{1}{\pi}\left|\det{Q^{-1}}\right|^{1/2}\exp(-y^{T}\cdot Q^{-1}\cdot y)\;,$

(24)

where

	$\displaystyle y^{T}$	$\displaystyle=$	$\displaystyle(r(t)-\langle r(t)\rangle,v(t)-\langle v(t)\rangle)\;,$
	$\displaystyle Q_{ij}$	$\displaystyle=$	$\displaystyle 2\langle y_{i}(t)y_{j}(t)\rangle\;.$		(25)

The averages in the above equations should be taken over an evolving swarm of trajectories that accounts for the initial conditions $r(0)=r_{0}=-a$ and $v(0)=v_{0}$. In terms of Laplace transformation of the generalized Langevin equation (3.1), the time evolution equations for $Q_{ij}$ can be calculated. Adelman showed that the probability distribution function $\rho(r,v,t)$ satisfies the generalized Fokker-Planck equation Adelman:1976 .

For our case, the free energy function near the top of the barrier $r=r_{m}$ can also be approximated as quadratic function as Eq.(3.2). Therefore, the generalized Fokker-Planck equation around the top of the potential barrier $r=r_{m}$ can be given by Adelman:1976

	$\displaystyle\frac{\partial}{\partial t}\rho(r,v,t)$	$\displaystyle=$	$\displaystyle\left[-v\frac{\partial}{\partial r}-\hat{\omega}_{m}^{2}r\frac{\partial}{\partial v}+\gamma(t)\left(\frac{\partial}{\partial v}v+k_{B}T\frac{\partial^{2}}{\partial v^{2}}\right)\right.$		(26)
			$\displaystyle\left.+k_{B}T\left(\frac{\hat{\omega}_{m}^{2}}{\omega_{m}^{2}}-1\right)\frac{\partial^{2}}{\partial r\partial v}\right]\rho(r,v,t)\;,$

with

	$\displaystyle\gamma(t)=-\frac{d}{dt}\ln\Phi(t)\;,$
	$\displaystyle\hat{\omega}_{m}^{2}(t)=-\theta(t)/\Phi(t)\;,$
	$\displaystyle\Phi(t)=\dot{\chi}(t)\left[1+\omega_{m}^{2}\int_{0}^{t}d\tau\chi(\tau)\right]-\omega_{m}^{2}\chi^{2}(t)\;,$
	$\displaystyle\theta(t)=\omega_{m}^{2}\left[\chi(t)\ddot{\chi}(t)-\dot{\chi}^{2}(t)\right]\;,$		(27)

and the function $\chi(t)$ is defined as the inverse Laplace transformation

$\displaystyle\chi(t)=\mathcal{L}^{-1}\left[\frac{1}{s^{2}-\omega_{m}^{2}+\hat{\zeta}(s)s}\right]\;.$

(28)

For a detailed derivation, one can refer to Adelman:1976 ; Hanggi:1982 . Comparing the Fokker-Planck equation for Markovian noise Li:2021vdp with the generalized Fokker-Planck equation (26), several similarities are remarkable. In Eq.(26), the last term is an additional diffusive term that is absent in the Markovian case. The other terms can also be found in the Markovian case. However, $\gamma(t)$ is time dependent while the corresponding coefficient in the Markovian case is a constant.

Next, one can find the approximate steady-state solution to the generalized Fokker-Planck equation (26). By inserting the ansatz Carmeli:1984

$\displaystyle\rho(r,v,t)=\frac{1}{Z}\Theta(r,v,t)e^{-\beta\left(v^{2}/2+G(r)\right)}\;,$

(29)

into the generalized Fokker-Planck equation (26), and assuming the function $\Theta(r,v,t)$ of the form

$\displaystyle\Theta(r,v,t)=\Theta(u,t)\;,\;\;u=v+\Gamma r\;,$

(30)

one can derive the resultant equation for $\Theta$ as given by

$\displaystyle\frac{\partial\Theta}{\partial t}=-k_{B}T(\bar{\lambda}+\Gamma)\frac{\partial^{2}\Theta}{\partial u^{2}}+\bar{\lambda}\left[v-\frac{\omega_{m}^{2}}{\bar{\lambda}}r\right]\frac{\partial\Theta}{\partial u}\;,$

(31)

where

$\displaystyle\bar{\lambda}=-\left[\gamma(t)+\Gamma\frac{\bar{\omega}_{m}^{2}(t)}{\omega_{m}^{2}}\right]\;.$

(32)

By solving the resultant equation with the appropriate boundary conditions, one can derive the steady-state solution of $\Theta$ as Schuller:2019ega

$\displaystyle\Theta(u)=\sqrt{\frac{\lambda}{2\pi A}}\int_{-\infty}^{u}\exp\left[-\frac{\lambda s^{2}}{2A}\right]ds\;,$

(33)

where

	$\displaystyle A=k_{B}T\left(\gamma-bc\right)\;,$
	$\displaystyle b=\frac{\omega_{m}^{2}}{\bar{\omega}_{m}^{2}}\left(\frac{\gamma}{2}+\sqrt{\frac{\gamma^{2}}{4}+\bar{\omega}_{m}^{2}}\right)\;,$
	$\displaystyle c=\frac{\bar{\omega}_{m}^{2}}{\omega_{m}^{2}}-1\;,$
	$\displaystyle\gamma=\lim_{t\rightarrow\infty}\gamma(t)\;,$
	$\displaystyle\bar{\omega}_{m}^{2}=\lim_{t\rightarrow\infty}\bar{\omega}_{m}^{2}(t)\;.$

and the reactive frequency $\lambda$ is determined by the equation Peters:2017

$\displaystyle\lambda=\frac{\omega_{m}^{2}}{\lambda+\hat{\zeta}(\lambda)}\;.$

(34)

Note that the frequency component of the time dependent friction $\hat{\zeta}(\lambda)$ is the inverse Laplace transformation of the friction kernel $\zeta(t)$

$\displaystyle\hat{\zeta}(\lambda)=\int_{0}^{\infty}e^{-\lambda t}\zeta(t)dt\;.$

(35)

The entire information about the friction kernel $\zeta(t)$ is therefore completely contained in $\lambda$. In practice, $\lambda$ is the largest root of the equation (34).

Finally, one can compute the transition rate from the small black hole state to the large black hole state, which is explicitly given by Grote:1980II

$\displaystyle k_{s\rightarrow l}=\frac{\lambda}{\omega_{m}}\frac{\omega_{s}}{2\pi}e^{-\beta W}\;,$

(36)

where $W=G(r_{m})-G(r_{s})$ is the barrier height between the small black hole state and the intermediate black hole state on the free energy potential for the RNAdS black holes. So the non-Markovian transition rate for the RNAdS black holes in the intermediate friction regime depends exponentially on the free energy barrier height between the two phases and also on the oscillating frequencies at the initial basin and barrier top as well as the reactive frequency.

Note that the MFPT is given by the inverse of the transition rate. Therefore, the MFPT in the intermediate friction regime is then given by

$\displaystyle\bar{t}=\frac{\omega_{m}}{\lambda}\frac{2\pi}{\omega_{s}}e^{\beta W}\;.$

(37)

It should be noted that the transition rate for the intermediate friction regime is also valid in the large friction limit. As a consistent check, one can show that the MFPT in the large friction limit Eq.(23) can be derived from the transition rate for the intermediate friction regime. In the large friction limit, from Eq.(34), one can see that $\lambda\rightarrow 0$, which in turn gives the approximate solution of $\lambda$ as

$\displaystyle\lambda=\frac{\omega_{m}^{2}}{\hat{\zeta}(0)}\;.$

(38)

Substituting the solution into the transition rate, one can get the MFPT formula in the large friction limit.

In this subsection, we consider the MFPT of the phase transition process in the weak friction regime. In the case of extremely weak friction, the dynamics is governed by energy diffusion or equivalently action diffusion because the energy change will be slow. In fact, Zwanzig Zwanzig:1959 derived an equation for the time evolution of the probability distribution function in the energy space with the non-Markovian effect Grote:1982 . It was later shown that Zwanzig’s diffusion equation corresponds to the generalized Langevin equation with the memory kernel represented by the time dependent friction and the random noise characterized by a finite correlation time Carmeli:1983 ; Carmeli:1982 . The main assumption used in Carmeli:1983 ; Carmeli:1982 to derive the diffusion equation in energy space is that the friction kernel will decay to zero in a finite time scale.

By introducing the action variable

$\displaystyle I(E)=\oint p(r)dr\;,\;\;p(r)=\sqrt{2(E-G(r))}\;,$

(39)

the probability distribution in the energy space satisfies the Fokker-Planck equation as Zwanzig:1959 ; Grote:1982 ; Carmeli:1983 ; Carmeli:1982

$\displaystyle\frac{\partial}{\partial t}\rho(I,t)=\frac{\partial}{\partial I}\left\{2\pi\epsilon(I)\left[2\pi k_{B}T\frac{\partial}{\partial I}+\omega(I)\right]\rho(I,t)\right\}\;,$

(40)

where the angular frequency $\omega(I)$ at the action $I$ and the diffusion coefficient in the energy space $\epsilon(I)$ are defined as

	$\displaystyle\omega(I)$	$\displaystyle=$	$\displaystyle 2\pi\frac{\partial E}{\partial I}\;,$
	$\displaystyle\epsilon(I)$	$\displaystyle=$	$\displaystyle\frac{1}{\omega^{2}(I)}\int_{0}^{\infty}\zeta(t)\langle v(0)v(t)\rangle dt\;.$		(41)

Hereby $v(t)$ should be obtained by solving the generalized Langevin equation without the dissipation or the friction kernel $\zeta(t)$ and the noise $\eta(t)$ for constant energy $E(I)$ and $\langle v(0)v(t)\rangle$ corresponds to averaging the initial phase.

From the diffusion equation in energy space, the MFPT to reach the final action $I$ from the initial action $I_{0}$ can be derived Grote:1982 ; Carmeli:1983 ; Carmeli:1982

$\displaystyle\hat{t}=\frac{1}{k_{B}T}\int_{I_{0}}^{I}\left\{\frac{1}{\epsilon(x)}e^{E(x)/k_{B}T}\int_{0}^{x}e^{-E(x^{\prime})/k_{B}T}dx^{\prime}\right\}dx\;.$

(42)

By performing the integral approximately, a compact approximate formula for the MFPT in the weak friction regime can be given as Grote:1982 ; Carmeli:1983 ; Carmeli:1982

$\displaystyle\bar{t}=\frac{k_{B}T}{\omega_{s}\epsilon(I_{m})\omega(I_{m})}e^{\beta W}\;,$

(43)

where $I_{m}$ is the action corresponding to the free energy $G(r_{m})$ and $\epsilon(I_{m})$ can be computed by using Eq.(3.4).

Using the parabola approximation of the generalized Gibbs free energy, one can further simplify the analytical expression (43) of the MFPT. Firstly, for the deterministic oscillatory movement inside the initial potential well, the action $I$ at the energy $G(r_{m})$ has the relation

$\displaystyle I_{m}=\frac{2\pi W}{\omega_{s}}\;.$

(44)

For the deterministic oscillation, one has

$\displaystyle\langle v(0)v(t)\rangle=\frac{W}{2}\left(e^{i\omega_{s}t}+e^{-i\omega_{s}t}\right)\;.$

(45)

Therefore, according to Eq.(3.4), $\epsilon(I_{m})$ can be given by

$\displaystyle\epsilon(I_{m})=\frac{W}{2\omega^{2}(I_{m})}\mathcal{F}[\zeta(t)](\omega_{s})\;,$

(46)

with $\mathcal{F}[\zeta(t)]$ being the Fourier transform of the friction kernel, which is defined as

$\displaystyle\mathcal{F}(\omega)=\int_{-\infty}^{+\infty}e^{i\omega t}\zeta(t)dt\;.$

(47)

Finally, the analytical expression of the MFPT in the weak friction regime is given by

$\displaystyle\bar{t}=\frac{2k_{B}T}{\mathcal{F}[\zeta(t)](\omega_{s})W}e^{\beta W}\;.$

(48)

In the large, intermediate, and weak friction regimes, the MFPTs are amenable to an analytical derivation. We have derived three analytical expressions of the MFPTs in different friction regimes, which are given by Eq.(23), (37), and (48). It should be noted that the expression for the MFPT in the large friction regime is included in the expression in the intermediate friction regime. Now we present a bridging formula of the MFPT that is valid in the whole friction regime Carmeli:1984 ; Schuller:2019ega ; Hanggi:1990 .

Note that the MFPT is proportional to the strength of the friction kernel in the intermediate-strong friction regime while it is inversely proportional to the friction strength in the weak friction regime. This is to say that $\bar{t}_{lowfriction}$ given by Eq.(48) will become shorter when the friction coefficient increases and $\bar{t}_{intermediatefriction}$ given by Eq.(37) will become shorter when the friction coefficient decreases, i.e., $\bar{t}_{lowfriction}$ and $\bar{t}_{intermediatefriction}$ have different behaviors with respect to the friction strength. Therefore, a very simple and intuitive approach to give the bridging formula that is valid in the whole friction regime is to interpolate between the already known formulas Carmeli:1984 ; Schuller:2019ega

$\displaystyle\bar{t}=\bar{t}_{lowfriction}+\bar{t}_{intermediatefriction}\;.$

(49)

When the friction coefficient approaches to $0$, $\bar{t}_{intermediatefriction}$ approaches $0$. When the friction coefficient approaches to infinity, $\bar{t}_{lowfriction}$ approaches $0$. Therefore, the kinetic time $\bar{t}$ given by the above formula is dominated by $\bar{t}_{lowfriction}$ in the low friction regime and by $\bar{t}_{intermediatefriction}$ in the large friction regime. The above formula gives the well behaved interpolation formula for the kinetic time in the whole friction regime. In the following sections, we will use this formula of the MFPT to investigate the non-Markovian dynamics of the Hawking-Page phase transition and the small/large RNAdS black hole phase transition.

The two analytical results of the MFPTs given by Eq.(48) and Eq.(23) imply that the kinetic time will be a decreasing function of friction in the small friction regime and an increasing function of the friction in the large friction regime. Therefore, in the intermediate friction regime, the kinetic turnover will be an universal property for any kinds of friction kernels.

Firstly, we consider the friction of the delta function

$\displaystyle\zeta(t)=\zeta\delta(t)\;.$

(50)

As mentioned, under this assumption, the non-Markovian Langevin equation reduces to the Markovian case. This assumption gives the Laplace transformation as

$\displaystyle\hat{\zeta}(\lambda)=\zeta\;,$

(51)

and

$\displaystyle\frac{\lambda}{\omega_{m}}=\left(\sqrt{\frac{\zeta^{2}}{4\omega_{m}^{2}}+1}-\frac{\zeta}{2\omega_{m}}\right)\;.$

(52)

Then the transition rate from one stable state to another stable state for the delta friction is given by

$\displaystyle k_{s\rightarrow l}=\left(\sqrt{1+\frac{\zeta^{2}}{4\omega_{m}^{2}}}-\frac{\zeta}{2\omega_{m}}\right)\frac{\omega_{s}}{2\pi}e^{-\beta W}\;.$

(53)

Note that this expression is only valid in the intermediate-large friction regime. For the small friction regime, we have

$\displaystyle\bar{t}=\frac{k_{B}T}{\zeta W}e^{\beta W}\;.$

(54)

This is consistent with the the result that derived in Li:2021vdp . There is a subtlety in deriving the above result when considering the boundary condition in deriving the MFPT for the weak friction. For small friction case, we explore the diffusion in the energy space. The first passage time is counted by the time that the initial state reaches the potential energy of the intermediate black hole state at the barrier top of the free energy landscape. Therefore, a factor of $1/2$ should be taken into account in the expression of the MFPT in the weak friction regime.

The two results given by (53) and (54) implies that the kinetic time will be a decreasing function of friction in the small friction regime and an increasing function of the friction in the large friction regime. Therefore, in the intermediate friction regime, there will be kinetic turnover. This reflects the fact that at high friction, the diffusion is along the order parameter $r$ and increasing friction leads to slower kinetics while at low friction, the diffusion dynamics is along the energy. Increasing the friction leads to larger perturbations to the original conserved system and this gives rise to faster the diffusion kinetics in energy space.

For the exponentially decayed friction, we mainly focus on the non-Markovian effects on kinetics of the black hole phase transition. In other words, we consider the effects of the time scale of the friction kernel. The time scale of the friction kernel describes the correlation time of the effective thermal bath of the underlying microscopic degrees of freedom for the black hole. In order to compare with the delta friction, i.e. Markovian dynamics, we consider the exponentially decayed friction which is given by

$\displaystyle\zeta(t)=\frac{\zeta}{\gamma}e^{-\frac{|t|}{\gamma}}\;,$

(55)

where $\zeta$ represents the strength of the friction and $\gamma$ is the time scale of the friction kernel. When $\gamma\rightarrow 0$, the exponentially decayed friction is reduced to the delta friction discussed in the last subsection.

For the exponentially decayed friction, the Laplace transformation is given by

$\displaystyle\hat{\zeta}(\lambda)=\frac{\zeta}{1+\lambda\gamma}\;.$

(56)

The equation that determines the prefactor $\lambda$ of the transition rate becomes a cubic equation from Eq.(34)

$\displaystyle\gamma\lambda^{3}+\lambda^{2}+\left(\zeta-\gamma\omega_{m}^{2}\right)\lambda-\omega_{m}^{2}=0\;.$

(57)

The $\lambda$ is given by the largest root of the above equation. By substituting the root into the analytical expression, one can calculate the MFPT of the phase transition process in the intermediate-strong friction regime.

For the small friction case, one should calculate the Fourier transformation of the friction kernel, which is given by

$\displaystyle\mathcal{F}(\omega)=\frac{2\zeta}{1+\omega^{2}\gamma^{2}}$

(58)

When $\gamma\rightarrow 0$, it gives $\mathcal{F}(\omega)=2\zeta$, which in consequence gives the MFPT in the weak friction regime for the delta friction. By taking the value of $\omega$ as the oscillating frequency of the initial potential well on the free energy landscape, one can get the MFPT of the transition process in the weak friction regime. It can be shown that $\mathcal{F}(\omega)$ is proportional to the friction coefficient $\zeta$, which implies that the MFPT in the weak friction regime is also inversely proportional to the strength coefficient of the friction kernel. Therefore, it is anticipated that there will also be a turnover point in the kinetics of the exponentially decayed friction when varying the friction coefficient.

In this subsection, we consider the final model of the time dependent friction. The friction kernel is oscillatory in time. For the oscillatory friction, we mainly consider the effects of the oscillating frequency on the kinetics of the black hole phase transition.

Our final model’s friction kernel is given by

$\displaystyle\zeta(t)=\zeta e^{-\frac{|t|}{\gamma}}\left[\cos(\tilde{\omega}t)+\frac{1}{\gamma\tilde{\omega}}\sin(\tilde{\omega}|t|)\right]\;,$

(59)

where $\zeta$ is the amplitude of the friction kernel denoting the strength of the friction, $\tilde{\omega}$ is the oscillating frequency of the friction kernel, and $\gamma$ is the decay coefficient or the time scale. It can be seen that the oscillatory friction kernel is the exponentially decayed friction plus the oscillating factor. In Figure 12, we have plotted the friction kernel of the oscillating friction for different decay coefficient $\gamma$. It can be observed that the friction kernel decays to zero in a finite time scale. So the analytical results are also valid for the oscillating friction kernel.

In order to calculate the MFPTs, we need the Laplace and Fourier transforms, which are given by

$\displaystyle\hat{\zeta}(\lambda)=\frac{\zeta\left(\lambda+2/\gamma\right)}{\tilde{\omega}^{2}+\left(\lambda+1/\gamma\right)^{2}}\;,$

(60)

and

$\displaystyle\mathcal{F}(\omega)=\frac{4\gamma\zeta\left(\tilde{\omega}^{2}\gamma^{2}+1\right)}{\left(\left(\tilde{\omega}-\omega\right)^{2}\gamma^{2}+1\right)\left(\left(\tilde{\omega}+\omega\right)^{2}\gamma^{2}+1\right)}\;.$

(61)

From the analytical expressions of the MFPTs together with the Laplace and Fourier transforms of the oscillatory friction, it can be easily observed that there will also be kinetic turnover points when plotting the relation between the MFPT and the friction coefficient. Another observation is that the Fourier transforms of the oscillatory friction attain a maximum when $\omega=\tilde{\omega}$ due to the term $\left(\left(\tilde{\omega}-\omega\right)^{2}\gamma^{2}+1\right)$, which implies that in the weak friction regime, there will be a resonance in the kinetics when the oscillating frequency is equal to the oscillating frequency for the initial black hole in the potential well on the free energy landscape. The resonance effect will be more significant for large $\gamma$. In the following, we will pay more attention to the resonance effect in the kinetics of the black hole phase transitions.