A quantum corrected Râ€"Nâ€"AdS black hole and it’s thermodynamics of phase transition.

To investigate the quantum gravity effects on the black hole thermodynamics, we studied the phase transition in a quantum corrected Râ€"Nâ€"AdS black holes. First, based on the work of Kazakov and Solodukhin, the spherically symmetric quantum fluctuation is applied to Râ€"Nâ€"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 â€" V diagram and phase transition for the quantum corrected Râ€"Nâ€"AdS black hole are studied. Compared with the existing research on Râ€"Nâ€"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 â€" 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 Râ€"Nâ€"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. [ABSTRACT FROM AUTHOR]