Joule-Thomson expansion of Lower-dimensional black hole

The Joule-Thomson expansion is extended to the lower-dimensional regime by considering the rotating BTZ metric in the (2+1)-dimensional space-time. Specifically, the properties of three important aspects of the Joule-Thomson expansion, namely the Joule-Thomson coefficient, the inversion curve and the isenthalpic curve are focused on. The divergence point of the Joule-Thomson coefficient and the zero point of the Hawking temperature are studied. The inversion temperature curves and isenthalpic curves in the $T-P$ plane are obtained and the cooling-heating regions are determined. Furthermore, the minimum inversion temperature is found to be zero, and the black hole becomes an extremal black hole. The ratio between the minimum inversion temperature and the critical temperature for the BTZ black hole doesn't exist, since the BTZ black hole does not have the critical behavior in $P_c$, $T_c$ and $V_c$.
Comment: 18 pages, 3 figures and 1 table, some minor changes