Thermodynamics and Joule-Thomson Expansion for Schwarzschild-AdS Black Holes with Cloud of Strings and Quintessential-like Fluid

TL;DR

This work analyzes a Schwarzschild–AdS black hole surrounded by a cloud of strings and a quintessence‑like fluid in extended phase space thermodynamics, treating the mass as enthalpy and the cosmological constant as pressure. It derives $M$, $T$, $S$, $F$, and $U$ with explicit dependence on the CoS $(|\alpha|, b)$ and QF $(N,w)$ parameters, and assesses stability via the specific heat $C_p$ and the Ruppeiner curvature $R_R$, linking thermodynamic and geometric signatures of phase structure. The inversion temperature and Joule–Thomson expansion are examined, showing that external fields shift inversion curves and cooling/heating regions, thus shaping the BH’s thermodynamic phase behavior. Overall, the study reveals rich thermodynamics and a clear geometric imprint of microphysical interactions, with large‑$r_+$ behavior approaching a universal, ideal‑gas‑like limit.

Abstract

In this study, we explore the thermodynamic properties of a Schwarzschild black hole (BH) embedded in an anti-de Sitter (AdS) background, which is further coupled with a cloud of strings and surrounded by a quintessence-like fluid. Beginning with the formulation of BH mass in terms of the event horizon radius, we incorporate the concept of pressure as related to the AdS curvature radius within the framework of extended phase space thermodynamics. Using this setup, we derive key thermodynamic quantities, including the Gibbs free energy and internal energy, to characterize the energetic behavior of the black hole system. To assess the stability of the black hole, we compute the specific heat capacity and analyze how it is influenced by external parameters, such as the string cloud and the quintessence-like fluid. These geometric and matter fields are shown to significantly modify the thermal response of the BH. Furthermore, we examine the inversion temperature associated with the black hole and highlight its distinction from the standard Hawking temperature, providing deeper insight into the phase structure. Additionally, we investigate the Joule-Thomson expansion process and demonstrate how the aforementioned parameters affect this thermodynamic phenomenon, showing important aspects of BH cooling and heating behavior in an extended thermodynamic context.

Figures (9)

• Figure 1: Behavior of the Hawking Temperature $T$ given in Eq. (\ref{['bb2']}) as a function of horizon for different values of CoS parameter $\alpha$ and state parameter $W$.
• Figure 2: Behavior of the Gibb's free energy $F$ given in Eq. (\ref{['F2']}) as a function of horizon for different values of CoS parameter $\alpha$ and state parameter $W$. Here, $\mathrm{N}=0.01,\,P=0.01$.
• Figure 3: Behavior of the internal energy $U$ given in Eq. (\ref{['F3']}) as a function of horizon for different values of CoS parameter $\alpha$ and state parameter $W$.
• Figure 4: Behavior of the specific heat capacity $C_p$ given in Eq. (\ref{['C4']}) as a function of horizon for different values of CoS parameters $\alpha$ and $b$. Here, we set $w=-2/3,\,\mathrm{N}=0.01,\,P=0.01$.
• Figure 5: Ruppeiner curvature scalar $R_R$ as a function of entropy $S$ for different values of $\alpha$ and fixed parameters $b=0.1$, $N=0.02$, $\omega=-2/3$, and $P=0.001$. The divergence points correspond to phase transitions identified via the specific heat analysis.