The P–v Criticality of a Noncommutative Geometry-Inspired Schwarzschild-AdS Black Hole

The P–v criticality and phase transition in the extended phase space of a noncommutative geometry-inspired Schwarzschild black hole in anti-de Sitter (AdS) spacetime are studied. The cosmological constant is treated as a dynamical pressure and its conjugate quantity is thermodynamic volume of the noncommutative geometry-inspired Schwarzschild-AdS black hole. The noncommutative parameter is also treated as a variable, and as a consequence, a new thermodynamic quantity V_\theta conjugate to P_\theta=-(8\pi \theta)^-1 has to be defined further, which is required for consistency of both the first law of thermodynamics and the corresponding Smarr relation. We find that the P–v criticality and the small black hole/large black hole phase transition appear for the noncommutative Schwarzschild-AdS black hole. Numerical calculations indicate that the noncommutative parameter \theta affects the phase transition as well as the critical temperature T_\rm c, horizon radius r_\rm +c and pressure P_\rm c. However, the critical ratio P_\rm cr_\rm +c/T_\rm c is universal (independent of \theta), which is very similar to the result in the van de Waals liquid–gas system, but different from that in the noncommutative geometry-inspired Reissner–Nordström-AdS black hole, where the critical ratio is no longer universal.