Thermodynamics of ($2+1$)-dimensional charged black holes with power-law Maxwell field

In this work, the three-dimensional nonlinearly charged black holes have been considered with a power-law modified electromagnetic theory. The black hole solutions to Einstein’s three-dimensional field equations with a negative cosmological constant have been constructed in the presence of power-law nonlinear electrodynamics. Through the physical and mathematical interpretation of the solutions, a new class of asymptotically anti–de Sitter (AdS) black hole solutions has been introduced. The area law, surface gravity, and Gauss’s law are utilized to obtain the entropy, temperature, and electric charge of the new AdS black holes, respectively. The quasilocal mass of the solutions has been calculated based on the counterterm method. A Smarr-type formula for the mass as a function of entropy and charge has been obtained. It has been shown that the thermodynamical quantities satisfy the first law of thermodynamics for the new AdS black holes. Also, it has been found that in order for the Smarr mass formula to be compatible with the first law of black hole thermodynamics, the cosmological parameter $\mathrm{\Lambda }$ should be treated as a thermodynamical variable and the generalized first law of thermodynamics has been introduced. Through the canonical ensemble method, the black hole remnant or phase transitions have been investigated regarding the black hole heat capacity. It has been found that the AdS black hole solutions we just obtained are thermodynamically stable.

Figure 1

$f(r)$ versus $r$ for $M=1$, $Q=1$, and $\ell =1$. Red, brown, and green lines correspond to $p=\frac{3}{5}$, $p=\frac{5}{7}$, and $p=\frac{9}{7}$, respectively. They show black holes with two horizons. The blue line ($p=\frac{4}{5}$) shows an extreme black hole, and the black line ($p=\frac{6}{7}$) shows the naked singularity. Note that Eqs. (3.16) and (3.24) have been used.