Thermodynamic nature of black holes in coexistence region

Studying the system state of coexistence regions will peek into to reveal microscopic interactions between different phases of a thermodynamic system.However,there is no effective method to study thermodynamic nature of the coexistence black hole regions for the failure of the equation of state.Aiming at these coexistence states,in this work,we develop a general approach by introducing two new ratio parameters.The first one is the ratio of the horizon radii of the saturated coexistence small and large black holes,and the second one measures that of the small black hole molecule number to the total molecule number.We demonstrate that the first parameter can serve as an order parameter to characterize the first-order phase transition.The study also shows that the black hole state in the coexistence region is uniquely determined by these two introduced parameters bounded between 0 and 1.These results are quite significant in the analytical study of phase transition and the microscopic nature of black hole in the coexistence regions.