Thermodynamics of charged rotating black strings in extended phase space

We investigate the thermodynamics of asymptotically Anti-de Sitter charged and rotating black strings in extended phase space, in which the cosmological constant is interpreted as thermodynamic pressure and the thermodynamic volume is defined as its conjugate. We find the thermodynamic volume, the internal energy, and the Smarr law. We study the thermal stability and show that some of the solutions have positive specific heat, which makes them thermodynamically stable. We find, for the first time, there is a critical point for charged solutions which occurs at the point of divergence of specific heat at constant pressure. This supports the existence of a second-order phase transition analogous to the liquid-gas critical point in Van der Waals fluids. We also study the maximal efficiency of a Penrose process and find that an extremal rotating black string can have an efficiency of up to 50%. We also find the equation of state for uncharged solutions. By comparing with the liquid-gas system, we observe that there is not a critical behavior to coincide with those of the Van der Waals system.