In this paper, we investigated the quantum gravity effects on the thermal properties of the [Formula: see text]-dimensional noncommutative rotating Banados–Teitelboim–Zanelli (NCR-BTZ) black hole in the context of quantum tunneling of relativistic particles. These include Hawking temperature, the thermally local and global stability conditions, and the phase transitions. For this purpose, in the framework of the generalized uncertainty principle (GUP), we used the Hamilton–Jacobi approach to calculate the tunneling probability for a massive scalar, Dirac, and vector boson particles from the [Formula: see text]-dimensional NCR-BTZ black hole. We found that the modified Hawking temperature of the black hole depends on the black hole properties, on the tunneling particle properties, on the noncommutative parameter, and on the GUP parameter. Using the modified Hawking temperature, we calculated the modified heat capacity, and then we discussed the local thermodynamic stability conditions for the black hole. The black hole may undergo a first-type phase transition to become stable under the scalar particle tunneling whereas, it might undergoes both the first and the second-type phase transitions under the both Dirac and vector boson particles tunneling process. Furthermore, we calculated the Gibbs free energy of the black hole, and we investigated the global stability conditions. We observed that Hawking–Page phase transition may occur in the presence of the quantum gravity effect under the tunneling process of scalar, Dirac, and vector boson particles. In the context of quantum gravity effect, we also derived the modified equation of state to investigate the critical behavior of the commutative rotating BTZ black hole. Finally, we shown that Van der Waals-like phase transition may occur in the context of tunneling process of both Dirac and vector boson particle, whereas it may not occur for the tunneling of scalar particle.

中文翻译：

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GUP 对非对易旋转 BTZ 黑洞热力学性质的影响

在本文中，我们研究了在相对论粒子的量子隧穿背景下，量子引力对 [公式：见正文] 维非对易旋转 Banados-Teitelboim-Zanelli (NCR-BTZ) 黑洞热性质的影响。这些包括霍金温度、局部和全局热稳定性条件以及相变。为此，在广义不确定性原理 (GUP) 的框架内，我们使用 Hamilton-Jacobi 方法从 [公式：见文本] 维 NCR 计算大质量标量、狄拉克和矢量玻色子粒子的隧穿概率-BTZ 黑洞。我们发现黑洞的修正霍金温度取决于黑洞属性、隧穿粒子属性、非对易参数和 GUP 参数。使用修正后的霍金温度，我们计算了修正后的热容量，然后讨论了黑洞的局部热力学稳定条件。黑洞在标量粒子隧穿过程中可能会经历第一类相变而变得稳定，而在狄拉克和矢量玻色子粒子隧穿过程下，它可能会同时经历第一类和第二类相变。此外，我们计算了黑洞的吉布斯自由能，并研究了全局稳定性条件。我们观察到在标量、狄拉克和矢量玻色子的隧穿过程下存在量子引力效应的情况下，可能会发生霍金-佩奇相变。在量子引力效应的背景下，我们还推导出了修正的状态方程来研究交换旋转 BTZ 黑洞的临界行为。最后，我们表明范德华相变可能发生在狄拉克和矢量玻色子粒子的隧穿过程中，而标量粒子的隧穿过程可能不会发生。