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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 $P = -\frac{\Lambda}{ 8 \pi}$. 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.