Critical behavior of charged dilaton black holes in AdS space

We revisit critical behavior and phase structure of charged anti--deSitter (AdS) dilaton black holes for arbitrary values of dilaton coupling $\ensuremath{\alpha}$, and realize several novel phase behavior of this system. We adopt the viewpoint that cosmological constant (pressure) is fixed and treat the charge of the black hole as a thermodynamical variable. We study critical behavior and phase structure by analyzing the phase diagrams in $T\ensuremath{-}S$ and $q--T$ planes. We numerically derive the critical point in terms of $\ensuremath{\alpha}$ and observe that for $\ensuremath{\alpha}=1$ and $\ensuremath{\alpha}\ensuremath{\ge}\sqrt{3}$, the system does not admit any critical point, while for $0<\ensuremath{\alpha}<1$, the critical quantities are not significantly affected by $\ensuremath{\alpha}$. We find that unstable behavior of the Helmholtz free energy for $q<{q}_{c}$ exhibits a first order (discontinuous) phase transition between small and large black holes for $0\ensuremath{\le}\ensuremath{\alpha}<1$, where ${q}_{c}$ is the value of charge at the critical point. For $1<\ensuremath{\alpha}<\sqrt{3}$ and $q>{q}_{c}$, however, a novel first order phase transition occurs between small and large black hole, which has not been observed in the previous studies on phase transition of charged AdS black holes.

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