Thermodynamic geometric analysis of 3D charged black holes under f(R) gravity

Abstract: This article investigates 3D charged black holes within the scope of f(R) gravity, focusing on their thermodynamic attributes. The research primarily examines minor fluctuations around these black holes' equilibrium states and delves into their modified thermodynamic entropy. Utilizing geometric thermodynamics (GTD), the study evaluates the curvature scalar's role in pinpointing phase transition points in these black holes. A key finding is that several 3D charged black holes under f(R) gravity display thermodynamic properties akin to an ideal gas when their initial curvature scalar remains constant. Conversely, with a non-constant curvature scalar and a cosmological constant term that includes a negative exponent, these black holes exhibit characteristics similar to a van der Waals gas. The article outlines general solutions for scenarios involving non-negative powers and specific solutions for cases with negative powers. Notably, under certain conditions, a phase transition resembling that of a van der Waals gas is observed, suggesting a strong correlation between the black hole's fate and the cosmological constant, extending beyond the parameters proposed by the no-hair theorem.The research provides insights into the swift decline of peaks linked to both large and small black holes, revealing new aspects of black hole transitional behaviors. In a three-dimensional space (for $d=3$) with a variable $k_1$ set to 1, and considering a $\Lambda$ term that adheres to SO(2) symmetry, the study uncovers a cusp catastrophe in the G-T function graph. This observation, within the specified metric, points to a distinct solution that characterizes the ``Phase Transition and Properties of Bose-Einstein Condensation" under specific conditions. Notably, this phase transition in Bose-Einstein condensation occurs due to the symmetry shift from SO(3) to SO(2).