Overview of black hole thermodynamics geometry and its microscopic structure

This paper presents a systematic review of research advancements on the microscopic molecular model of black holes, with a central focus on elucidating the intrinsic connection between black hole thermodynamic phase transitions and their underlying microscopic structures. By integrating thermodynamic principles with statistical mechanical frameworks, the study aims to bridge macroscopic observations with microscopic interactions, offering insights into the fundamental nature of black hole gravity. The discussion begins with an in-depth analysis of the small-large black hole phase transition. We detail the rigorous methodologies for identifying critical points and characterizing the associated critical phenomena, such as power-law scaling of thermodynamic quantities near the transition. These results serve as the foundational basis for constructing a microscopic model that can quantitatively reproduce macroscopic phase transitions.Building on the universality principles of phase transitions, we propose a foundational framework for the microscopic molecular model of black holes. This model posits that black hole thermodynamics may emerge from the statistical behavior of hypothetical “microscopic constituents" analogous to molecules in a fluid, with their interactions dictating macroscopic observables. To further explore this connection, we introduce Ruppeiner geometry within the thermodynamic parameter space, analyzing both its line element and scalar curvature. A critical comparative analysis is then conducted between the Ruppeiner geometries of black holes and the classical van der Waals fluid. While both systems exhibit phase transitions, their geometric features diverge significantly: the black hole system demonstrates unique curvature signatures indicative of repulsive interactions. These differences highlight the specialized microscopic structure of black holes, driven by their inherently gravitational nature. Guided by these insights, we reconstruct plausible forms of the interaction potential between black hole microscopic constituents, accounting for the nontrivial statistical behaviors revealed by the geometric analysis. Ultimately, this work synthesizes the logical pathway from phase transition phenomena to microscopic model construction. By doing so, it advances our understanding of the gravitational nature of black holes.