本文利用York形式研究了正则系综中五维Schwarzschild黑洞的热力学。在一个大小固定、温度固定的$r$和$T$的空腔内，在$πr T = 1$处有一个阈值，超过这个阈值，黑洞可以处于热平衡。对于两个特定的黑洞，一个视界半径为$r_{+1}$的小黑洞和一个半径为$r_{+2}$的大黑洞，可以实现这种热平衡。在五个维度中，半径$r_{+1}$和$r_{+2}$具有精确的表达式。通过路径积分形式和配分函数，得到作用量和自由能。这导致了系统的热能和熵，后者由贝肯斯坦-霍金面积定律$S = \frac{A_{+}}{4}$给出，其中$A_+$是黑洞的表面积。当热浴放置在等于或小于光子轨道的半径$r$处时，热容为正，这意味着热力学稳定性。这意味着小黑洞是不稳定的，而大黑洞是稳定的。利用广义自由能证明了经典热平坦空间通过$r_{+1}$到达稳定的$r_{+2}$是可行的.值得注意的是，当空腔半径等于布赫达尔半径时，较大的$r_{+2}$黑洞的自由能为零。的不稳定性所产生的扰动，由于在瞬子的解决方案中的路径积分的关系被提及。量子热平坦空间具有负自由能，我们找到了大黑洞、量子热平坦空间或两者都是基态的条件。显示相应的相图。在给定能量$E$的情况下，利用态密度$ν$，我们还发现大黑洞$r_{+2}$的熵。此外，我们还将五维热力学与York的四维结果联系起来。
英文摘要：
We study the thermodynamics of a five-dimensional Schwarzschild black hole in the canonical ensemble using York's formalism. Inside a cavity of fixed size $r$ and fixed temperature $T$, there is a threshold at $πr T = 1$ above which a black hole can be in thermal equilibrium. This thermal equilibrium can be achieved for two specific black holes, a small black hole of horizon radius $r_{+1}$, and a large black hole of radius $r_{+2}$. In five dimensions, the radii $r_{+1}$ and $r_{+2}$ have an exact expression. Through the path integral formalism and the partition function, one obtains the action and the free energy. This leads to the thermal energy and entropy of the system, the latter turning out to be given by the Bekenstein-Hawking area law $S = \frac{A_{+}}{4}$, where $A_+$ is the black hole's surface area. The heat capacity is positive when the heat bath is placed at a radius $r$ that is equal or less than the photonic orbit, implying thermodynamic stability. This means that the small black hole is unstable and the large one is stable. A generalized free energy is used to show that it is feasible that classical hot flat space transits through $r_{+1}$ to settle at the stable $r_{+2}$. Remarkably, the free energy of the larger $r_{+2}$ black hole is zero when the cavity radius is equal to the Buchdahl radius. The relation to the instabilities that arise due to perturbations in the path integral in the instanton solution is mentioned. Quantum hot flat space has negative free energy and we find the conditions for which the large black hole, quantum hot flat space, or both are the ground state. The corresponding phase diagram is displayed. Using the density of states $ν$ at a given energy $E$ we also find that the entropy of the large black hole $r_{+2}$. In addition, we make the connection between the five-dimensional thermodynamics and York's four-dimensional results.
Analytic semiclassical backreaction of a Schwarzschild black hole in a finite cavity: horizon shift, temperature renormalization, and canonical stability in the Hartle-Hawking State
Centre for Theoretical Physics, The British University（英国大学理论物理中心）；Department of Mathematics and Statistics, Imam Mohammad Ibn Saud Islamic University (IMSIU)（伊马姆穆罕默德·本·沙特伊斯兰大学数学与统计系）；Faculty of Physics, Babeş-Bolyai University（巴比什-博雅依大学物理学院）；Astronomical Observatory（天文台）
The enriched phase structure of black branes in canonical ensemble
研究发现，腔中D维带电黑p膜（$p > 0$）的相结构的完成需要正则系综中的两个额外的非线性相，即所谓的“空泡”和/或极值膜.这一发现解决了当$\tilde d = D - p - 3 \leq 2$和给出一个新的（气泡）阶段，可以成为全球稳定时，$\tilde d > 2$的基础相图所需的丢失的阶段的难题。在其它新的相变中还发现了类似Hawking-Page相变的相变，给出了该体系的完整相结构。
