The Study Of Thermodynamic Phase Transitions And Microstructures Of Black Holes

In this article,we introduce the origin and development of black hole thermodynamics,the introduction of extended phase space,and then use equal area law and Ruppenier geometry to study the thermodynamic phase transition properties and microstructure of black holes,respectively.So far,research on the thermodynamic properties of black holes has mainly gone through two stages.The first stage involved completing the transformation from the four laws of black hole mechanics into the four laws of black hole thermodynamics,which opened the prelude to the study of black hole thermodynamics.The second stage involves extending the first law of black hole thermodynamics by interpreting the negative cosmological constant as pressure,thus opening a new era of research on the thermodynamic properties of black holes in extended phase space.An important tool for studying the phase transition behavior of black holes in extended phase space is the equal area law.For the first time,we extend the equal area law to the case of two-valued functions,and the equal area law of single-valued functions is a special case of the equal area law of two-valued functions.We first calculated the analytic small-large black hole coexistence curve of the five-dimensional charged Gauss-Bonnet Anti-de Sitter(GB-Ad S)black hole in the grand canonical ensemble by using the equal area law of the single-value function.Using this coexistence curve,we studied the phase diagram of the P-V plane and the critical behavior of the system.We then study the low-high potential phase transition of the four-dimensional charged Ad S black hole by using the equal area law of the two-valued function.Using different expressions of the equal area law at places close to the critical point(two-valued function)and far away from the critical point(single-valued function)of the system,we obtain the coexistence curve of low-high potential black holes in both cases.Using the coexistence curve,we studied the phase diagram of the Q-Φ plane in detail,constructed the order parameters,and calculated the critical exponent near the critical point.We also introduced the Ruppeiner geometry,applied the Ruppeiner geometry to the five-dimensional charged GB-Ad S black hole,and calculated the normalized scalar curvature of the system.Combing with the empirical observation of scalar curvature,we find that for low electric potential,the attractive interaction dominates among the microstructures,while a high electric potential produces repulsive interactions.In the reduced parameter space,we observe that only attractive interaction is allowed when the coexistence region is excluded.The normalized scalar curvature also admits a critical exponent 2 and a universal constant 1/8.In particular,the value of the normalized scalar curvature keeps the same along the coexistence small and large black hole curves.So in the grand canonical ensemble,the interaction can keep constant at the phase transition where the black hole microstructures change.These results disclose the intriguing microstructures for the charged Ad S black hole in the Gauss-Bonnet gravity.