In this paper, we consider the solutions of Einstein gravity in the presence of a generalized Maxwell theory, namely power Maxwell invariant. First, we investigate the analogy of nonlinear charged black hole solutions with the Van der Waals liquid–gas system in the extended phase space where the cosmological constant appear as pressure. Then, we plot isotherm $P-V$ diagram and study the thermodynamics of AdS black hole in the (grand canonical) canonical ensemble in which (potential) charge is fixed at infinity. Interestingly, we find the phase transition occurs in the both of canonical and grand canonical ensembles in contrast to Reissner-Nordström black hole in Maxwell theory which only admits canonical ensemble phase transition. Moreover, we calculate the critical exponents and find their values are the same as those in mean field theory. Besides, we find in the grand canonical ensembles universal ratio $\frac{P_c{v}_c}{T_c}$ is independent of spacetime dimensions.

• Received 11 August 2013

DOI:https://doi.org/10.1103/PhysRevD.88.084045