P–v criticality in the extended phase space of a noncommutative geometry inspired Reissner–Nordström black hole in AdS space-time

The P–v criticality and phase transition in the extended phase space of a noncommutative geometry inspired Reissner–Nordstrom (RN) black hole in Anti-de Sitter (AdS) space-time are studied, where the cosmological constant appears as a dynamical pressure and its conjugate quantity is thermodynamic volume of the black hole. It is found that the P–v criticality and the small black hole/large black hole phase transition appear for the noncommutative RN-AdS black hole. Numerical calculations indicate that the noncommutative parameter affects the phase transition as well as the critical temperature, horizon radius, pressure and ratio. The critical ratio is no longer universal, which is different from the result in the van de Waals liquid–gas system. The nature of phase transition at the critical point is also discussed. Especially, for the noncommutative geometry inspired RN-AdS black hole, a new thermodynamic quantity \(\varPsi \) conjugate to the noncommutative parameter \(\theta \) has to be defined further, which is required for consistency of both the first law of thermodynamics and the corresponding Smarr relation.