Effects of Gauss-Bonnet term on the phase transition of a Reissner-Nordström-AdS black hole in ($3+1$) dimensions

In this paper, we investigate the phase transition of Reissner-Nordström-AdS (RN-AdS) black holes in ($3+1$)-dimensional Gauss-Bonnet gravity in the extended thermodynamic phase space. Since the Gauss-Bonnet term is a topological invariant, it does not contribute to the spacetime, the global charges and their conjugate potentials, the equation of state, and the critical phenomena macroscopically or the coexistence states, coexistence curves, and molecule number densities microscopically. However, the entropy, the first law of thermodynamics, and Smarr relation are modified owing to the Gauss-Bonnet term. Also, in contrast to the case in Einstein gravity, the Hawking-Page phase transition happens at a different temperature induced by the modified entropy. Especially, the positivity of the entropy results in a constraint on the special volume, which does bring the RN-AdS black hole to the reentrant phase transition and changes the whole phase structure. The temperatures of the zero and first order phase transition are analytically given. The phase structure is dependent on the electric charge $Q$ and Gauss-Bonnet coupling $\alpha$, while in the reduced parameter space, it depends only on the dimensionless special volume of zero entropy state ${\nu }_S=\frac{\sqrt{2|\alpha |}}{\sqrt{3}Q}$, rather than $Q$ and $\alpha$. Moreover, this is the first analytical example in which the structure of the reentrant phase transition and the effect of the entropy constraint are presented.

Figure 1

Curve of the ratio $\frac{T_0}{T_0^E}$ for the cases with general $\mathrm{\Xi }$, which characterizes the effect of the Gauss-Bonnet term on the Hawking-Page temperature. The red point denotes the Hawking-Page temperature in Einstein gravity. When $\alpha >0$, the phase transition temperature becomes smaller, while it grows when $\alpha <0$.

Figure 2

Curves of $T-v$ and $G-T$ for a RN-AdS black hole with $P<P_c$. The thick lines denote the globally stable states of the system. There are the $T-v$ oscillatory behavior and the classical “swallow tail” characterizing the small/large ($v_1/v_2$) black hole phase transition.

Figure 3

Curves of $G-T$ for a RN-AdS black hole in Gauss-Bonnet gravity. The phase structure depends on the zero entropy state S. The left panel corresponds to the case without a phase transition because half of the swallow tail is absent. The middle panel corresponds to the case with the first order phase transition, as there still exists the classical swallow tail characterizing the small/large (${\nu }_1/{\nu }_2$) black hole phase transition. The right panel corresponds to the case with the reentrant phase transition. The thick lines denote the globally stable states of the systems, while the dotted lines denote the unphysical states with negative entropy.