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Ehrenfest's scheme and microstructure for regular-AdS black hole in the extended phase space

The regular (Bardeen)-AdS (BAdS) black hole (BH) in the extended phase space is taken as an example for investigating the BH phase transition grade from both macroscopic and microscopic points of view. The equation of state and thermodynamic quantities of this BH are obtained. It is found that the BAdS BH phase space in the extended phase space should be a second-order phase transition near the critical point by verifying the Ehrenfest's equation, and the possibility of its first-order phase transition can be ruled out by the entropy continuity and the heat capacity mutation. The critical exponents from the microscopic structure are analytically and numerically presented with the Landau continuous phase transition theory by introducing a microscopic order-parameter.