Thermodynamics and the Joule-Thomson expansion of dilaton black holes in 2+1 dimensions

TL;DR

This work analyzes static charged dilaton black holes in 2+1 dimensions within the extended phase space, treating the cosmological constant as pressure and promoting the dilaton-related parameters $\beta$ and $\gamma$ to thermodynamic status to close the first law. The authors derive the Smarr relation, compute the temperature, entropy, thermodynamic volume, electric potential, and Gibbs free energy, and study local and global stability across two regimes: $\frac{2}{3}\le N<1$ (small black holes locally stable, large ones not) and $1\le N<2$ (globally stable). They find no Hawking–Page or Van der Waals-type phase transitions for the values examined and analyze the Joule–Thomson expansion, showing inversion curves and isenthalpic behavior, with the inversion structure depending on $N$ (including two inversion curves for $N=\tfrac{2}{3}$). The paper also tests the Reverse Isoperimetric Inequality, showing parameter regions where the inequality holds due to the dilaton field and regions where superentropic behavior is possible; comparisons with BTZ black holes highlight the distinctive role of the dilaton in 2+1 dimensions.

Abstract

In this paper, we study thermodynamics and its applications of a family of static charged dilaton black holes in 2+1 dimensions found by Chan and Mann \cite{Chan:1994qa} and Xu \cite{Xu:2019pap}. There is a dimensionless parameter $N$ in the black hole solutions presented: it is related to the coupling constant for the dialton with the electromagnetic field and the gravitational field. Black hole horizons exists only for $ \frac{2}{3} \leq N < 2$. $N =1$ black hole is a solution to low energy string theory. Thermodynamics are studied in the canonical ensemble where charge is constant. The cosmological constant is considered a thermodynamical variable where the pressure $P = -\fracΛ{ 8 π}$. We computed the first law and the Smarr relations for the black hole and introduced two new thermodynamical parameters in order to satisfy the first law. We computed temperature, thermodynamic volume, specific heat capacities, Gibbs free energy and studied local and global stability of the black hole. Thermodynamic volume differs from the geometric volume. We noticed that thermodynamic behavior falls into two broad categories: For $\frac{2}{3} \leq N < 1$, small black holes are locally stable and large black holes are not. For $ 1 \leq N < 2$ the black hole is locally and globally stable for all values of the horizon radius. In order to demonstrate the two broad categories, we have presented $N =1, \frac{2}{3}$ and $N = \frac{6}{7}$ black holes in detail. There were no phase transitions for the above values of $N$. We have also studied the Joule-Thomson expansion and the Reverse Isoperimetric Inequality of these black holes. We made the observation that the charged dilaton black hole does not violate the Reverse Isoperimetric Inequality for certain values of the parameters of the theory. Finally, we have suggested future work.

Figures (29)

• Figure 1: The figure shows the temperature $T$ vs the horizon radius $r_+$ for varying values of $N$. Here $Q = 1,\beta = 12.88, \Lambda = -0.562$.
• Figure 2: The figure shows the temperature $C_{P,Q}$ vs the horizon radius $r_+$ for varying values of $N$. Left graph is for $N < 1$ and the right graph is for $N > 1$. Here $\Lambda = -0.1, \gamma = 1$ for both graphs. For the left graph, $Q = 0.5, \beta = 17$ and for the right graph $Q = 0.386, \beta = 0.5$
• Figure 3: The figure shows $f(r)$ vs $r$ for the dilaton black hole for $N = \frac{6}{7}$. Here $Q = 0.376, \beta = 0.1, \gamma = 1, \Lambda = -1$. The blue curve corresponds $M = 0.144$ and is a black hole solution with two horizons. The red curve corresponds to $M = 0.104$ and does not have horizons. The green curve corresponds to an extreme black hole with $M = 0.118$.
• Figure 4: The figure shows the pressure $P$ vs the horizon radius $r_+$ for fixed charge. Here $N = 6/7, Q = 0.259, T = 0.406$ and $\beta = 0.1$.
• Figure 5: The figure shows the specific heat capacity $C_{P,Q}$ and Hawking temperature $T$ vs the horizon $r_+$ for fixed charge. Here $N = 6/7, Q = 0.5$, $\Lambda = -1, \gamma =1$, and $\beta =$ 0.97, 0.644, 0.486 (green, red and blue).