To investigate the quantum gravity effects on the black hole thermodynamics, we studied the phase transition in a quantum corrected R–N–AdS black holes. First, based on the work of Kazakov and Solodukhin, the spherically symmetric quantum fluctuation is applied to R–N–AdS black hole, and the metric is given. Next, we investigate the basic thermodynamic quantity and Smarr formula, and then propose that the quantum correction parameter a should also be treated as a variable to ensure the establishment of the new Smarr formula. Finally, the P–V diagram and phase transition for the quantum corrected R–N–AdS black hole are studied. Compared with the existing research on R–N–AdS black hole, some effects by quantum fluctuation are analyzed and presented. First, the quantum correction will weaken the charge effect of the black hole. Second, the P–V curve of the quantum corrected black hole is no longer analogous to that of Van Der Waals (VDW) gas exactly. Third, the black hole has two more phase states than R–N–AdS black hole. One is a remnant phase state, the other is an unstable phase state. And both are derived from the quantum corrections of the black hole.

中文翻译：

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量子校正的 R–N–AdS 黑洞及其相变热力学

为了研究量子引力对黑洞热力学的影响，我们研究了量子校正 R–N–AdS 黑洞中的相变。首先，基于Kazakov和Solodukhin的工作，将球对称量子涨落应用于R-N-AdS黑洞，并给出度量。接下来，我们研究基本的热力学量和 Smarr 公式，然后提出量子校正参数A也应将其视为一个变量，以确保新 Smarr 公式的建立。最后，P–V研究了量子校正 R-N-AdS 黑洞的图和相变。与现有的 R-N-AdS 黑洞研究相比，分析并提出了量子涨落的一些影响。首先，量子校正会削弱黑洞的电荷效应。第二，P–V量子校正黑洞的曲线不再与范德华 (VDW) 气体的曲线完全相似。第三，黑洞比 R–N–AdS 黑洞多了两个相态。一种是残相状态，另一种是不稳定相状态。并且两者都是从黑洞的量子校正中推导出来的。