Phase transition and thermodynamic geometry of Einstein-Maxwell-dilaton black holes

In this paper, we consider linearly charged dilatonic black holes and study their thermodynamical behavior in the context of phase transitions and thermodynamic geometry. We show that, depending on the values of the parameters, these black holes can undergo two types of phase transition. We also find that there are three critical behaviors near the critical points for these black holes: nonphysical unstable to physical stable, large to small, and small to large black-hole phase transitions. Next, we employ a thermodynamical metric to study the thermodynamical geometry of these black holes. We show that the characteristic behavior of the Ricci scalar of this metric enables one to recognize the type of phase transition and critical behavior of the black holes near phase-transition points. Finally, we will extend thermodynamical space by considering the dilaton parameter as an extensive parameter. With this consideration, we will show that the Weinhold, Ruppeiner, and Quevedo metrics provide extra divergencies that are not related to any phase-transition point, whereas our new method provides an effective mechanism to study phase transitions.