P-v criticality and heat engine efficiency for Bardeen Einstein-Gauss-Bonnet AdS black hole

In this paper, we discuss the P–v criticality and the heat engine efficiency in the Bardeen Einstein–Gauss–Bonnet (EGB) AdS black hole space–time. From the P–v plane in the extended phase space, we find that the Bardeen EGB-AdS black hole conforms to Van der Waals (VdW) liquid–gas systems in the extended phase space, and $\tfrac{{P}_{c}{v}_{c}}{{T}_{c}}=0.369$ of the Bardeen EGB-AdS black hole system is between 0.3333 of the Gauss–Bonnet AdS black hole system and 0.375 of the VdW gas system in the 5-dimensions. Then we construct a heat engine by taking the Bardeen EGB-AdS black hole as the working substance, and consider a rectangle heat cycle in the P–v plane. We find that two cases with different Bardeen parameter e and Gauss–Bonnet parameter α both have the same situation, i.e. as the entropy difference between small black hole S₁ and large black hole S₂ increases, the heat engine efficiency will increase. Furthermore, as the Bardeen parameter e increases, the efficiency will decrease. However, for the Gauss–Bonnet parameter α, the result is contrary. By comparing with the well-know Carnot heat engine efficiency, we have found the efficiency ratio η/η_c versus entropy S₂ is bounded below 1, so it is coincided with the thermodynamical second law.