在反德西特时空下， 将宇宙学常数Λ 视为压强P， 其共轭量为热力学体积V。引入了P、V 两个物理量后，黑洞热力学四定律的内容得到了丰富。2012 年，Kubiznak 和Mann 比较分析了RN AdS黑洞的小大黑洞之间的相变与范德瓦尔斯流体气液相变。后来，人们在不同维数的黑洞解不仅发现了范德瓦尔斯相变，还发现了三相点、重入相变等丰富的相变类型。

本学位论文工作主要做了两方面的工作。一方面是应用光子球的特征参量来反映Power Maxwell AdS黑洞相变， 另一方面是应用福克-普朗克方程讨论分析了dRGT AdS黑洞相切换的热动力学。

首先，在第二章我们讨论了Power Maxwell AdS黑洞解的热力学相变与光子球轨道之间的关系。我们发现光子球轨道与最小碰撞参数的等压线和等温线的行为与黑洞热力学是一致的，存在着振荡行为。我们可以利用这一非单调行为来表征小大黑洞相变。以s =为例，通过数值计算，我们得到小大黑洞共存的P-T 曲线。我们讨论了光子球轨道与最小碰撞参数沿共存曲线的变化。通过数值拟合，我们发现，在临界点附近，与 逐渐为0，并且光子球轨道半径r_ps 和最小碰撞参数u_ps 的临界指数都是。这强烈地暗示着光子球可以反映黑洞热力学相变。

然后，在第三章中，我们应用福克- 普朗克方程讨论了dRGT AdS 黑洞在四维情况下的范德瓦尔斯相变。我们发现黑洞的初始相并不影响确定压强与温度下黑洞相的概率密度分布。接着，我们在局域不稳定黑洞相处设定吸收边界条件来讨论黑洞相切换这一过程。我们得到了初始状态为小黑洞或大黑洞相时的首通时间分布F_p(t)，由此我们也可以得到这一过程平均值，我们称为平均首通时间< t >。通过计算讨论，我们可以看到，当压强小于临界压强时，初始位置在小黑洞相时，随着温度升高，F_p(t) 峰值升高并且左移，< t > 减小，说明当小黑洞相向大黑洞相切换的热动力学过程加快，小黑洞相更容易向大黑洞相切换；而大黑洞相向小黑洞相切换时则与之相反。除此之外，我们还发现， 当小黑洞相向大黑洞相切换时，这一行为在高温情况下，首通时间分布更靠近平均首通时间，并且温度T 相较势垒高度△G_L 对小黑洞相切换的动力学影响也更显著。而大黑洞相向小黑洞相相切换时，则与之相反。

为了更近一步讨论黑洞相变热动力学的情况，在第四章中，我们讨论了六维情况下，dRGT AdS黑洞解的三相点和类范德瓦尔斯相变的热动力学。在这两类相变类型中仍保持着初始状态不影响最终的概率密度平稳分布。在讨论三相点这一相变类型时，我们发现当初始位置在中黑洞相时，势垒宽度∆r 对黑洞相变的动力学也有影响。势垒宽度越宽，黑洞相转变的首通时间分布的峰值就越低，极值点与平均首通时间就越大，说明黑洞相切换就越困难。并且，通过对三相点的中黑洞相讨论，我们发现中黑洞相在实现相切换时存在着优先方向。本文首次讨论了类范德瓦尔斯相变情况下的黑洞相切换的热动力学。在类范德瓦尔斯相变情况下，小黑洞和大黑洞相实现相的切换时，与d = 4 时范德瓦尔斯情况下的小黑洞相和大黑洞相相切的变化一致，仍是温度的单调函数。唯一的不同之处在于类范德瓦斯相变存在中黑洞相。我们发现，中黑洞相的各个物理量不再是温度T 的单调函数。接着我们计算了中黑洞相两端的势垒高度，发现两端的势垒高度仍保持着对温度的单调性，只是增减性不同。由此我们推断，温度通过影响中黑洞相两端的势垒高度来影响首通时间分布F_p(t)、平均首通时间< t >，从而影响中黑洞相相切换的方向选择。

最后，我们在第五章对前面的讨论进行了总结，并对后续可能开展的工作进行了归纳。

Under the anti-de sitter spacetime, the cosmological constant Λ seems as pressure P and then its conjugate quantity is the thermodynamics volume V. Introducing these two terms P and V, the contents of the black hole thermodynamics law are completed. In 2012, Kubiznak and Mann compared the two phases of RN AdS black holes with van der Waals fluid phase change of gas and liquid. Later, people found not only Van der Waals phase transition but also other rich phase transition types, such as triple point and reentrant phase transitions in different solutions of black holes.

This paper has done two aspects.  On the one hand,we use the characteristic parameter of the photon sphere to reflect the transition of Power Maxwell AdS black hole. On the other hand, we use the Fokker-Planck equation to discuss the phase switching thermodynamics of dRGT AdS black hole.

We first discusses the relationship of Power Maxwell AdS black holes between thermody- namic phase transition and photon sphere orbit in the chapter 2. We find that the isobar and isotherm of photon orbit and minimum impact parameters are in keeping with black hole ther- modynamic, and exist oscillatory behaviors.  We  can use these non-monotone behaviors to characterize the phase transition between small black hole and big black hole.We take s =   as the example.By numerical computation, we can get the P − T coexistence curve when small black hole and large black hole can coexist. We discuss the photon orbit’s and the minimum impact parameters’ behaviors along the coexistence curve. By numerical fitting, we find that near the critical point,andgradually tend to zero. What’s more, the critical number of sphere orbit r_ps and the minimum impact parameter u_ps are both ½.It strongly implies that the of photon sphere can reflect the thermodynamic phase transition of the black hole.

Then,in the chapter 3, we apply the Fokker-Planck equation to discuss dRGT AdS black holes’ Van der Waals phase transition in the four dimensional case. We find that the initial condition doesn’t influence the distributions of probability in the specific pressure and temper- ature.Then, we set the Absorbing boundary condition at the unstable black hole phase, in order  to discuss the process of black hole phases switching.We get the first passage process F_p(t) when the initial state is small black hole or large black hole, so that we also can get the mean first pas- sage time < t > and corresponding fluctuation of these two processes.  By calculation,  we can  see that in the pressure which is less than the critical pressure when the initial state is the small black hole, the peak of F_p(t) will get higher and move left, and < t > will decrease as the temper- aturebeing higher. It means that the switching process of small black hole to large black hole is more quickly and easily, while the switching process of large black hole to small black hole is on the contrary.What’s more.We find that the first passage process of the small black hole switching to the big black hole is more closed to the mean first passage time in the higher temperature,and the temperature has the more influence to the small black holes’ thermal dynamics compared with the barrier height △G_L, while the behavior of the big black is opposite to it.

In order to further discuss the thermal dynamics of black hole phase transitions, in Chapter 4,we discuss the thermodynamics of triple point and Van der Waals type phase transitions of dRGT AdS black hole solutions in the six-dimensional case. In these two phase transition types,the initial condition still have no influence on the final stationary distributions of probability.When discussing the triple point, we find that when the initial state is at the intermediate black hole, The barrier width also has influence on the black hole thermal dynamic phase transition.When the barrier width is more larger,the peak of F_p(t) will be lower, and the extreme point and the the mean first passage time will be longer. That means the process of black hole switching is more difficult. In addition, by discussing the intermediate black hole phase of the triple point, we find that the middle black hole phase has a preferential direction when implementing the switch. In this passage we first disscuss the thermodynamic phase transition of black hole in the situation of VdW type phase transition. In the type of VdW type phase transition, the phase switching process of small and large black hole are same as VdW phase transition in the case of 4 dimension.But there is only one difference in the VdW type phase transition. It is that the VdW type phase transition has the intermediate black hole which is also the stable black hole phase.We find that the individual quantities of the intermediate black hole phase are no longer monotone functions of temperature T . And then we calculate the intermediate black hole’s barrier height, it is found that the height of the barrier at both ends still maintains the monotonicity of the temperature, but the one is increasing ,the another one is decreasing. It is inferred that the specific temperature affects the first passage time F_p(t) and the mean first pass time < t > by affecting the height of the barrier at both ends of the intermediate black hole phase, thus affecting the phase switching direction of the intermediate black hole.

In the end, the previous discussion is summarized in the chapter 5 and summarize the possible follow-up work.