Dynamic property of phase transition for non-linear charged anti-de Sitter black holes ^*

The action for four-dimensional Einstein-power–Yang–Mills (EPYM) gravity with a cosmological constant Λ is given by [70–73]

$$I = \frac{1}{2}\int {\rm d}^4x\sqrt{g} \left(R-2\Lambda-[{\rm Tr}(F^{(a)}_{{\mu\nu}}F^{{(a)\mu\nu}})]^\gamma\right) \tag{ 1 }$$

with the Yang–Mills (YM) field

$$F_{\mu \nu}^{(a)} = \partial_{\mu} A_{\nu}^{(a)}-\partial_{\nu} A_{\mu}^{(a)}+\frac{1}{2 \xi} C_{(b)(c)}^{(a)} A_{\mu}^{b} A_{\nu}^{c}, \tag{ 2 }$$

where ${\rm Tr}(F^{(a)}_{\mu\nu}F^{(a)\mu\nu}) = \sum^3_{a = 1}F^{(a)}_{\mu\nu}F^{(a)\mu\nu}$, R is the scalar curvature, γ is a positive real parameter, $C_{(b)(c)}^{(a)}$ represents the structure constants of three-parameter Lie group G, ξ is the coupling constant, and $A_{\mu}^{(a)}$ are the $SO(3)$ gauge group YM potentials.

For this system, the black hole solution of the corresponding field equation with the negative cosmological constant Λ is [74]:

$${\rm d} s^{2} = -f(r) {\rm d} t^{2}+f^{-1} {\rm d} r^{2}+r^{2} {\rm d} \Omega_{2}^{2}, \tag{ 3 }$$

$$f(r) = 1-\frac{2 M}{r}-\frac{\Lambda}{3} r^{2}+\frac{\left(2 q^{2}\right)^{\gamma}}{2(4 \gamma-3) r^{4 \gamma-2}}, \tag{ 4 }$$

where ${\rm d}\Omega_{2}^{2}$ is the metric on unit $2$-sphere with volume $4\pi$, and q is the YM charge. Note that this solution is valid for the condition where the non-linear YM charge parameter $\gamma\neq0.75$ and the power YM term holds the weak energy condition (WEC) for $\gamma \gt, 0$ [71]. In the extended phase space, Λ was interpreted as the thermodynamic pressure $P = -\dfrac{\Lambda}{8\pi}$. The black hole event horizon locates at $f(r_+) = 0$. The parameter M represents the ADM mass of the black hole and reads [10]

$$\begin{aligned}[b] H = & M(S, q, P) = \frac{1}{6}\left[8 \pi P\left(\frac{S}{\pi}\right)^{3 / 2}\right.\\ & \left.+3\left(\frac{S}{\pi}\right)^{({3-4 \gamma})/{2}} \frac{\left(2 q^{2}\right)^{\gamma}}{8 \gamma-6}+3 \sqrt{\frac{S}{\pi}}\right]. \end{aligned} \tag{ 5 }$$

In our set up, this parameter is associated with the enthalpy of the system. The black hole temperature, entropy, and volume are given by [70]

$$\begin{aligned}[b] T = & \frac{1}{4 \pi r_+}\left(1+8 \pi P r_+^{2}-\frac{\left(2 q^{2}\right)^{\gamma}}{2 r_+^{(4 \gamma-2)}}\right),\\ S = & \pi r_+^{2},\; \; \; \; V = \frac{4\pi r_+^3}{3}. \end{aligned} \tag{ 6 }$$

The YM potential Ψ is given by [64]

$$\Psi = \frac{\partial M}{\partial q^{2 \gamma}} = \frac{r_+^{3-4 \gamma} 2^{\gamma-2}}{4\gamma-3}. \tag{ 7 }$$

The above thermodynamic quantities satisfy the first law

$${\rm d} M = T {\rm d} S+\Psi {\rm d} q^{2 \gamma}+V {\rm d} P. \tag{ 8 }$$

The equation of state $P(V,T)$ for the canonical ensemble (fixed YM charge q) can be obtained from the expression of temperature as

$$P = \left(\frac{4 \pi}{3 V}\right)^{1 / 3}\left[\frac{T}{2}-\frac{1}{8 \pi}\left(\frac{4 \pi}{3 V}\right)^{1 / 3}+\frac{\left(2 q^{2}\right)^{\gamma}}{16 \pi}\left(\frac{4 \pi}{3 V}\right)^{({1-4\gamma})/{3}}\right]. \tag{ 9 }$$

From Eq. (9), we know the state equation of the EPYM black hole with fixed YM charge corresponds to the one of an ordinary thermodynamic system and can be written as $f(T,P,V) = 0$. Furthermore, the number of particles in the system is unchanged. Through Maxwell's equal-area law, we can construct the phase transition of the EPYM black hole in $P-V$, $T-S$, and $q^{2\gamma}-\Psi$, respectively. For the same parameters, the phase transition points in $P-V$, $T-S$, and $q^{2\gamma}-\Psi$ should be the same. In the following we will present the phase diagram in $T-S$ and label the phase transition point with $T_0$ and $P_0$.

For the EPYM black hole with a given YM charge q and pressure $P_0 \lt P_{\rm c}$, the entropy values at the boundary of the two-phase coexistence area are $S_1$ and $S_2$, respectively. The corresponding temperature is $T_0$, which is less than the critical temperature $T_{\rm c}$ and is determined by the horizon radius $r_+$. Therefore, from the Maxwell's equal-area law $T_0(S_2-S_1) = \int^{S_2}_{S_1}T{\rm d}S$ and Eq. (6), we have

$$\begin{aligned}[b] 2 \pi T_{0} = & \frac{1}{r_{2}(1+x)}+\frac{8 \pi P_0 r_{2}}{3(1+x)}\left(1+x+x^{2}\right)\\ & -\frac{\left(2 q^{2}\right)^{\gamma} r_{2}^{1-4 \gamma}}{2(3-4 \gamma)} \frac{\left(1-x^{3-4 \gamma}\right)}{\left(1-x^{2}\right)} \end{aligned} \tag{ 10 }$$

with $x = \dfrac{r_1}{r_2}$. In addition, from the state equation we have

$$T_{0} = \frac{1}{4 \pi r_{2}}\left(1+8 \pi P_0 r_{2}^{2}-\frac{\left(2 q^{2}\right)^{\gamma}}{2 r_{2}^{(4 \gamma-2)}}\right),\; \; \; \; \tag{ 11 }$$

$$T_{0} = \frac{1}{4 \pi r_{1}}\left(1+8 \pi P_0 r_{1}^{2}-\frac{\left(2 q^{2}\right)^{\gamma}}{2 r_{1}^{(4 \gamma-2)}}\right). \tag{ 12 }$$

From Eq. (10), we have

$$0 = -\frac{1-x}{r_{2} x}+8 \pi P_0r_{2}(1-x)+\frac{\left(2 q^{2}\right)^{\gamma}}{2 r_{2}^{4 \gamma-1} x^{4 \gamma-1}}\left(1-x^{4 \gamma-1}\right), \; \; \; \; \tag{ 13 }$$

$$8 \pi T_{0} = \frac{1+x}{r_{2} x}+8 \pi P_0r_{2}(1+x)-\frac{\left(2 q^{2}\right)^{\gamma}}{2 r_{2}^{4 \gamma-1} x^{4 \gamma-1}}\left(1+x^{4 \gamma-1}\right). \tag{ 14 }$$

Considering Eqs. (10), (13), and (14), we find the larger horizon has the following form:

$$\begin{aligned}[b] r_{2}^{4 \gamma-2} = & \frac{\left(2 q^{2}\right)^{\gamma}\left[(3-4 \gamma)(1+x)\left(1-x^{4 \gamma}\right)+8 \gamma x^{2}\left(1-x^{4 \gamma-3}\right)\right]}{2 x^{4 \gamma-2}(3-4 \gamma)(1-x)^{3}} \\= & \left(2 q^{2}\right)^{\gamma} f(x,\gamma). \end{aligned} \tag{ 15 }$$

Since the above state parameters must be positive, the non-linear YM charge parameter satisfies the condition $\dfrac{1}{2} \lt \gamma$. For the critical point ( $x = 1$), the state parameters are

$$r_{\rm c}^{4 \gamma-2} = \left(2 q^{2}\right)^{\gamma} f(1, \gamma), \quad f(1, \gamma) = \gamma(4 \gamma-1), \tag{ 16 }$$

$$T_{\rm c} = \frac{1}{\pi\left(2 q^{2}\right)^{\gamma /(4 \gamma-2)} f^{1 /(4 \gamma-2)}(1, \gamma)} \frac{2 \gamma-1}{4 \gamma-1}, \tag{ 17 }$$

$$P_{\rm c} = \frac{2 \gamma-1}{16 \pi \gamma\left(2 q^{2}\right)^{\gamma /(2 \gamma-1)} f^{1 /(2 \gamma-1)}(1, \gamma)}. \tag{ 18 }$$

Substituting Eq. (15) into Eq. (13), we have the following expression:

$$\begin{aligned}[b] \frac{1-x}{x} = & 8\pi P_0 \left(2 q^{2}\right)^{{\gamma}/({2 \gamma-1})}f^{{1}/({2 \gamma-1})}(x,\gamma)\\ & +\frac{1-x^{4 \gamma-1}}{2x^{4 \gamma-1}f(x,\gamma)}f^{({4 \gamma-3})/({4 \gamma-2})}(x,\gamma).\end{aligned} \tag{ 19 }$$

For a given parameter γ and pressure $P_0$, we can obtain the value of x from the above equation. Then from Eq. (15), we know that for a given pressure $P_0$ ( $P_0 \lt P_{\rm c}$), i.e., for a fixed value of x, the phase transition condition reads

$$\frac{\left(2 q^{2}\right)^{\gamma}}{r_{2}^{4 \gamma-2}} = \frac{1}{f(x, \gamma)}. \tag{ 20 }$$

Therefore, the phase transition of the EPYM black hole with a given temperature $P_0$ ( $P_0 \lt P_{\rm c}$) is determined by the ratio between the YM charge $(2q^2)^\gamma$ and $r_{2}^{4 \gamma-2}$, not only the value of the horizon. Note that we call this ratio the YM electric potential with the horizon radius $r_2$. Therefore, the phase transition of this thermodynamic system is the high-/low-potential black hole (HPBH/LPBL) phase transition. The plot of the phase transition in the $T-S$ diagram with fixed pressure $P_0 = 0.85955P_{\rm c}$ is shown in Fig. 1. The effects of the non-linear parameter γ and YM charge q on phase transition were exhibited in our last work [40]

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For the EPYM charged AdS black hole with non-linear charge, we proposed the existence of the HPBH/LPHB phase transitions in Ref. [40], not only the large/small black holes transition. In the following we present the thermal dynamic phase transition at the phase transition point ( $T_0 = 0.0402$ and $P_0 = 0.85955P_{\rm c}$) with the parameters $q = 0.85,\; \gamma = 0.8$.

The Gibbs free energy landscape of the charged EPYM AdS black hole reads

$$G_{\rm L} = M-T_{E}S, \tag{ 21 }$$

where $T_E$ is a temperature parameter and equals $T_0$, which is not the Hawking temperature. The picture of the Gibbs free energy landscape at the phase transition point of $P_0 = 0.85955P_{\rm c}$ and $T_E = T_0 = 0.0402$ is shown in Fig. 2. From this picture, we can see that, at the phase transition point, the Gibbs free energy landscape displays double-well behavior. Namely there are two local minima (located at $r_l = 1.4577,\; r_s = 3.0904$) which correspond to the stable high-/low-potential black hole states with positive heat capacity. The local maximum located at $r_m = 2.2107$ represents the unstable intermediate-potential black hole state with negative heat capacity, and acts as a barrier between the stable HPBH and LPBH states. Furthermore, the depths of two local minima are the same. This indicates that the HPBH/LPBH phase transition will occur at the same depth for the two wells from the view of Gibbs free energy landscape. At this point, we expect that the reentrant phase transition or triple point will maybe correspond to more wells of the Gibbs free energy landscape.

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As shown in the previous section, we find that the HPBH/LPBH phase transition of the charged EPYM AdS black hole emerges when the double wells of $G_{\rm L}$ have the same depth. Next we will investigate the dynamic process of the HPBH/LPBH phase transition for the charged EPYM AdS black hole.

Recently, authors proposed that the stochastic dynamic process of black hole phase transition can be studied by the associated probabilistic Fokker–Planck equation on the Gibbs free energy landscape [50], which is an equation of motion governing the distribution function of flucturating macroscopic variables. For a black hole thermodynamic system, the horizon $r_+$ is the order parameter and can be regarded as a stochastic fluctuating variable during the phase transition. Based on this, we will exhibit the dynamical process of the HPBH/LPBH phase transition in the canonical ensemble under thermal fluctuations. Note that the canonical ensemble consists of a series of black holes with arbitrary horizons. The probability distribution of these black hole states $\rho(t,r_+)$ satisfies the Fokker–Planck equation on the Gibbs free energy landscape:

$$\frac{\partial\rho(t,r_+)}{\partial t} = D\frac{\partial}{\partial r_+}\left({\rm e}^{-\beta G_{\rm L}(r_+)}\frac{\partial}{\partial r_+}\left[{\rm e}^{\beta G_{\rm L}(r_+)}\rho(t,r_+)\right]\right), \tag{ 22 }$$

where $\beta = 1/kT_E$, $D = kT_E/\xi$ is the diffusion coefficient with k being the Boltzman constant and ξ being the dissipation coefficient. Without loss of generality, we set $k = \xi = 1$. In order to solve the above equation, two types of boundary ( $r_+ = r_{0}$) condition should be imposed. One is the reflection boundary condition, which preserves the normalization of the probability distribution. The other is the absorption boundary condition.

In this system the location of the left boundary should be smaller than $r_s$, and the right boundary is larger than $r_l$. Since the temperature of the charged EPYM AdS black hole with pressure $P_0 = 0.85955P_{\rm c}$ and parameters $q = 0.85,\; \gamma = 0.8$ should be not negative, there exists a minimum value $r_{\rm min} = 0.695808$. We can regard $r_{\rm min}$ as the left boundary $r_{\rm lb}$, and set the right boundary $r_{\rm rb}$ as $6$. The reflection boundary condition means the probability current vanishes at the left and right boundaries:

$$j(t,r_0) = -T_E{\rm e}^{-G_{\rm L}/T_E}\frac{\partial}{\partial r_+}\left({\rm e}^{G_{\rm L}/T_E}\rho(t,r_+)\right)\mid_{r_+ = r_0} = 0 . \tag{ 23 }$$

The absorption condition means the probability distribution function vanishes at the boundary: $\rho(t,r_0) = 0$. The adoption of this boundary condition is determined by considering the physical problem.

The initial condition is chosen as a Gaussian wave packet located at $r_i$:

$$\rho(0,r_+) = \frac{1}{\sqrt{\pi}}{\rm e}^{-{(r_+-r_i)^2}/{a^2}}. \tag{ 24 }$$

Here a is a constant which determines the initial width of the Gaussian wave packet, and its value does not influence the final result. Since we mainly consider the thermal dynamic phase transition between HPBH/LPBH states, $r_i$ can be set to $r_s$ or $r_l$. This means this thermal system is initially in the high- or low-potential black hole state.

The time evolution of the probability distribution is shown in Fig. 3. At $t = 0$ the Gaussian wave packets locate at LPBH (Fig. 3(a)) and HPBH (Fig. 3(b)) with $T_E = T_0$ and $a = 0.1$, respectively. They both decrease with increasing time t until tending to a certain constant. However, at the same time the peaks of $\rho(t,r)$ at $r = r_{s}$ (Fig. 3(a)) and $r = r_{s}$ (Fig. 3(b)) are increasing from zero to the same constant. This indicates that the black hole system in the LPBH phase tends to the HPBH phase as shown Fig. 3(a), while in the HPBH phase tends to the low phase as shown Fig. 3(b). Finally the system reaches a LPBH/HPBH coexistence stationary state after a short time. In order to further clarify the dynamical process of the HPBH/LPBH phase transition, we show the probability distributions of $\rho(t,r_l)$ and $\rho(t,r_s)$ in Fig. 4. The probability distribution of the initial LPBH (or HPBH) is at its maximum, whereas the corresponding HPBH (or LPBH) state is at zero. With increasing time, both approach to the same value. This is consistent with what was shown in $G_{\rm L}-r_+$, i.e., the LPBH and HPBH states have the same depth in the double well of the Gibbs free energy landscape.

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In general, an important quantity in the dynamical process of the HPBH/LPBH phase transition is the first passage time, which is defined as the mean value of the first passage time that a stable HPBH or LPBH scape to the unstable intermediate-potential black hole state (i.e., from the one well to the barrier of Gibbs free energy landscape).

Suppose there is a perfect absorber in the stable HPBH or LPBH state. If the system makes the first passage under thermal fluctuation, then the system will leave this state. We can define Σ to be the summed probability of the dynamical process within the first passage time as

$$\Sigma = \int^{r_{\rm m}}_{r_{\rm min}}\rho(t,r_+){\rm d}r_+, \; \; \; \; \; \; {\rm{or}}\; \; \; \; \; \Sigma = -\int^{r_{\rm m}}_{r_{\rm rb}}\rho(t,r_+){\rm d}r_+, \tag{ 25 }$$

where $r_{\rm m}$, $r_{\rm min}$, $r_{\rm rb}$ are the intermediate, minimum, and right boundary of charged EPYM AdS black hole horizons. After a long time, the probability of the stable HPBH or LPBH state still in this system becomes zero, i.e., $\Sigma(t,r_l)\mid_{t\rightarrow \infty} = 0$ or $\Sigma(t,r_s)\mid_{t\rightarrow \infty} = 0$. As claimed, the first passage time is a random variable because the dynamical process of phase transition is caused by thermal fluctuation. Hence, we denote the distribution of the first passage time by $F_p$, which reads

$$F_p = -\frac{{\rm d}\Sigma}{{\rm d}t}. \tag{ 26 }$$

It is obviously that $F_p{\rm d}t$ indicates the probability of the system passing through the intermediate-potential black hole state for the first passage in the time interval ( $t,\; t+{\rm d}t$). Considering Eqs. (22) and (25), the distribution of the first passage time $F_p$ becomes [52]

$$F_p = -D\frac{\partial\rho(t,r_+)}{\partial r}{\bigg|}_{r_{\rm m}}, \; \; \; \; \; {\rm or}\; \; \; \; F_p = D\frac{\partial\rho(t,r_+)}{\partial r}{\bigg|}_{r_{\rm m}}. \tag{ 27 }$$

Here the absorbing and reflecting boundary conditions of the Fokker–Planck equation are imposed at $r_{\rm m}$ and the other limit ( $r_{\rm min}$ or $r_{\rm rb}$). Note that the normalisation of the probability distribution is not preserved.

By solving the Fokker–Planck Eq. (22) with different phase transition temperatures ( $T_0 = 0.0402$ and $T_0 =$ $0.038$) and substituting these into Eqs. (25) and (26), numerical results are obtained, as displayed in Figs. 5 and 6 for different initial conditions. It is clear that in Fig. 5, no matter what kind of initial condition has been considered, Σ decays very quickly. Furthermore, the increase of temperature makes the probability of Σ drop faster. An important point is that the probability is not conserved. From the corresponding probability distribution picture in Fig. 6, the behavior of $F_p$ is similar for both initial conditions. A single peak emerges near $t = 0$ in the curve of $F_p$ with fixed temperature. This can be understood as a large number of first passage events occurring in a short interval of time, and then the probability distribution decays exponentially with time. The effect of temperature at phase transition points on $F_p$ is consistent with that of Σ and $G_{\rm L}$. That means that the higher the temperature, the faster the probability decreases, the easier the phase transition occurs, and the lower the depth of the barrier. Conversely, the lower the temperature, the slower the probability decreases, the harder the phase transition, and the higher the depth of the barrier.

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