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 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 alpha both have the same situation, i.e. as the entropy difference between small black hole S-1 and large black hole S-2 increases, the heat engine efficiency will increase. Furthermore, as the Bardeen parameter e increases, the efficiency will decrease. However, for the Gauss-Bonnet parameter alpha, the result is contrary. By comparing with the well-know Carnot heat engine efficiency, we have found the efficiency ratio eta/eta(c) versus entropy S-2 is bounded below 1, so it is coincided with the thermodynamical second law.