Responses of AdS Black Holes to the Collective Influence of Quintessence and String Cloud

This research paper explores the interplay between quintessence and a cloud of strings, focusing on their influence on critical points, behaviors, and fractional-order phase transitions in AdS black holes. We analyze the thermodynamic properties of AdS black holes surrounded by quintessence and a cloud of strings, with particular attention to the effects of thermal fluctuations on their thermodynamics. Using critical conditions, we derive approximate analytical expressions for the critical points applicable to all types of these black holes. We further study the impact of the quintessence parameter and the intensity of the cloud of strings on critical behaviors, employing three-dimensional visualizations to depict these effects for different values of omega q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _q$$\end{document}. Our results show how these factors affect the locations of phase transitions and the regions where both phases coexist. In addition, we examine fractional-order phase transitions, which reveal significant transformations in the black hole system.