Thermodynamics of ( 2 + 1 )-dimensional black holes in Einstein-Maxwell-dilaton gravity

In this paper, the linearly charged three-dimensional Einstein’s theory coupled to a dilatonic field has been considered. It has been shown that the dilatonic potential must be considered in a form of generalized Liouville-type potential. Two new classes of charged dilatonic black hole solutions, as the exact solutions to the Einstein-Maxwell-dilaton (EMd) gravity, have been obtained and their properties have been studied. The conserved charge and mass related to both of the new EMd black holes have been calculated. Through comparison of the thermodynamical extensive quantities (i.e., temperature and entropy) obtained from both, the geometrical and the thermodynamical methods, the validity of first law of black hole thermodynamics has been investigated for both of the new black holes we just obtained. At the final stage, making use of the canonical ensemble method and regarding the black hole heat capacity, the thermal stability or phase transition of the new black hole solutions have been analyzed. It has been shown that there is a specific range for the horizon radius in such a way that the black holes with the horizon radius in that range are locally stable. Otherwise, they are unstable and may undergo type one or type two phase transitions to be stabilized.