Department of Mathematics

In this thesis, we investigate the thermodynamic properties of the Bardeen-Kiselev black hole solution with a cosmological constant, focusing on stability, phase transitions, and the compressibility factor using generalized entropy corrections. By incorporating Bar row, Renyi, Sharma-Mittal, Kaniadakis, and loop quantum gravity entropy corrections, we ´ explore their impact on black hole stability and phase transitions through specific heat ca pacity, pressure, compressibility, and Helmholtz free energy. The analysis reveals that spe cific heat capacity exhibits distinct behaviors across entropy models, including regions of stability (positive specific heat), instability (negative specific heat), and divergence points indicating phase transitions. For instance, Barrow entropy shows stability for small black holes (BHs), phase transitions for intermediate BHs, and restabilization for larger BHs. Loop quantum gravity entropy highlights significant phase transitions with unique stability patterns at higher horizon radii. Pressure-volume graphs illustrate how entropy corrections influence the black hole’s equation of state. Smaller BHs often exhibit instability at low volumes, transitioning to stable phases as volume increases. Notably, the Barrow and Sharma-Mittal entropy models show smooth transitions, whereas Kaniadakis and loop quantum gravity entropy correc tions reveal more abrupt changes. The compressibility factor analyses further underscore the non-ideal thermodynamic behavior of black holes under these entropy corrections. While Barrow entropy maintains compressibility near zero, Kaniadakis and loop quan tum gravity corrections exhibit exponential divergences, highlighting significant deviations from classical thermodynamics. Finally, Helmholtz free energy graphs demonstrate entropy-corrected phase transitions ix and stability trends. Smaller BHs exhibit consistent phase shifts across all models, with larger BHs progressively stabilizing under increasing pressure. Comparative analysis indi cates that Barrow entropy provides higher stability for larger BHs, while Sharma-Mittal and loop quantum gravity corrections lead to more dynamic transitions. This comprehensive study enriches the understanding of black hole thermodynamics by bridging classical and quantum gravity perspectives. By interpreting the cosmological constant as pressure and incorporating entropy corrections, this work offers insights into black hole stability, phase transitions, and their connections to statistical mechanics and quantum gravity frameworks. These findings contribute to the broader understanding of black hole behavior in general relativity and beyond.

Department of Mathematics

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