$P\text{−}V$ criticality of charged dilatonic black holes

In this paper, we investigate the critical behavior of charged black holes of Einstein-Maxwell-dilaton gravity in the presence of two Liouville-type potentials which make the solution asymptotically neither flat nor AdS and has a parameter $\mathrm{\Lambda }$ treated as a thermodynamic quantity that can vary. We obtain a Smarr-type relation for charged dilatonic black holes and find out that the volume is different from the geometrical volume. We study the analogy of the Van der Waals liquid-gas system with the charged dilatonic black hole system, while we treat the black hole charge as a fixed external parameter. Moreover, we show that the critical values for pressure, temperature, and volume are physical provided the coupling constant of dilaton gravity is less than one and the horizon is a sphere. Finally, we calculate the critical exponents and show that they are universal and are independent of the details of the system although the thermodynamic quantities depend on the dilaton parameter and the dimension of the spacetime.

Figure 1

$P-v$ diagram of charged dilatonic black hole in the case of $k=1$ for $n=3$, $q=b=1.0$, $\alpha =0$ (left) and $\alpha =0.3$ (right).

Figure 2

$P-v$ diagram of charged dilatonic black hole in the case of $k=1$ for $n=4$, $q=b=1.0$, $\alpha =0.3$ (left) and $\alpha =0.5$ (right).

Figure 3

Gibbs free energy of charged dilatonic black hole with $\alpha =0.3$, $q=b=1.0$ for $n=3$ (left) and $n=4$ (right).

Figure 4

Coexistence curves of the small-large black hole phase transition of a charged dilatonic black hole for $q=b=1.0$ and different values of $\alpha$ with $n=3$ (left) and $n=4$ (right) in the P-T plane. The critical points are highlighted by a small circle at the end of the coexistence curves.