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

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 dilaton with the electromagnetic field and the gravitational field. Black hole horizons exist only for $ \frac{2}{3} \leq N < 2$. $N =1$ black hole is a solution to low energy string theory. Thermodynamics is studied in the canonical ensemble where charge is constant as well as in grand canonical ensemble where the potential is constant. The cosmological constant is considered as a thermodynamical variable where the pressure $P = -\frac{\Lambda}{ 8 \pi}$. We computed the first law for the black hole and introduced new thermodynamical parameter 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. In the canonical ensemble, 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 $N$. In the grand canonical ensemble, we noticed that there is a Hawking-Page phase transition for the black hole with $N=6/5$. We have also studied the Joule-Thomson expansion and the Reverse Isoperimetric Inequality of these black holes...