Thermal Analysis, Joule-Thomson Expansion and Hawking Sparsity of Mod(A)Max-AdS Black Hole Immersed in a Cloud of Strings

Abstract

We investigate the thermodynamic behavior of a spherically symmetric Anti-de Sitter black hole in Mod(A)Max electrodynamics surrounded by a cloud of strings. Within the extended phase-space framework, we treat the cosmological constant as a pressure and interpret the black-hole mass as enthalpy, which enables a unified discussion of local stability, global phase structure, and Joule--Thomson expansion. We analyze the Hawking temperature, Gibbs free energy, and heat capacity, and show how the string-cloud parameter, the Mod(A)Max deformation, and the electric charge reshape the physical domain, the stability windows, and the small/large black-hole transition pattern. We further characterize the critical behavior and demonstrate that a van der Waals--like phase structure arises only in the physical sector, while the alternate branch does not admit a genuine critical point. For the Joule-Thomson process, we determine the inversion curve and the corresponding isenthalpic trajectories, highlighting how the model parameters control the cooling/heating regimes and can generate terminating isenthalpic behavior at sufficiently large charge. Finally, we examine the sparsity of Hawking radiation and discuss how the underlying parameters influence the temporal discreteness of the emitted flux, particularly near extremality and in the large-radius AdS regime.

Figures (6)

• Figure 1: Hawking temperature as a function of the horizon radius for the Mod(A)Max--AdS black hole surrounded by a cloud of strings. In panels (a) and (b) we plot $T(r_h)$ for fixed $(\gamma,\alpha,P)=(0.5,0.1,0.01)$ and several charge values $Q=\{0.3,0.5,0.7,0.9,1.1\}$, for $\eta=+1$ and $\eta=-1$, respectively. Panels (c) and (d) display $T(r_h)$ for fixed $(Q,\alpha,P)=(0.7,0.1,0.01)$ and $\gamma=\{0.0,0.3,0.6,0.9,1.2\}$, again for $\eta=+1$ and $\eta=-1$, respectively. The zeros of $T$ separate physical ($T>0$) and nonphysical ($T<0$) branches and determine the extremal limit.
• Figure 2: Gibbs free energy as a function of the Hawking temperature for the Mod(A)Max--AdS black hole surrounded by a cloud of strings. Panels (a) and (b) display $G(T)$ for fixed $(Q,\gamma,\alpha)=(0.7,0.5,0.1)$ and several pressures $P=\{0.005,0.008,0.010,0.012,0.015\}$, for $\eta=+1$ and $\eta=-1$, respectively. Panels (c) and (d) show $G(T)$ for fixed $(Q,\gamma,P)=(0.7,0.5,0.01)$ and distinct cloud-of-strings parameters $\alpha=\{0.0,0.1,0.2,0.3,0.4\}$, again for $\eta=+1$ and $\eta=-1$, respectively. The swallowtail structure (when present) signals a first-order small/large black-hole phase transition, while its disappearance indicates the approach to the critical behavior.
• Figure 3: Heat capacity at constant pressure as a function of the horizon radius for the Mod(A)Max--AdS black hole surrounded by a cloud of strings. Panels (a) and (b) show $C_{P}(r_h)$ for fixed $(\gamma,\alpha,P)=(0.5,0.1,0.01)$ and several charge values $Q=\{0.3,0.5,0.7,0.9,1.1\}$, for $\eta=+1$ and $\eta=-1$, respectively. Panels (c) and (d) display $C_{P}(r_h)$ for fixed $(Q,\alpha,P)=(0.7,0.1,0.01)$ and $\gamma=\{0.0,0.3,0.6,0.9,1.2\}$, again for $\eta=+1$ and $\eta=-1$, respectively. The sign of $C_{P}$ distinguishes locally stable ($C_{P}>0$) from unstable ($C_{P}<0$) branches, while divergences indicate second-order transition points associated with changes in local stability.
• Figure 4: Equation of state and criticality structure in the extended phase space for the Mod$(A)$ black hole with a cloud of strings. The specific volume is identified as $v=2r_h$. Panels (a) and (b) display the isotherms $P(v)$ for fixed $\alpha=0.1$, $\gamma=0.5$, and $Q=0.7$, using reduced temperatures $T/T_c=\{0.80,0.90,1.00,1.10\}$ (computed from the $\eta=+1$ critical point). Panel (a) corresponds to $\eta=+1$ and exhibits a van der Waals--like behavior with the critical point $(v_c,P_c)$ marked. Panel (b) corresponds to $\eta=-1$, for which the isotherms do not develop a physical inflection-point criticality. Panel (c) shows the universal reduced equation of state in terms of $P_r=P/P_c$, $T_r=T/T_c$, and $v_r=v/v_c$ for $\eta=+1$, $P_r=\frac{8T_r}{3v_r}-\frac{2}{v_r^{2}}+\frac{1}{3v_r^{4}},$ highlighting the mean-field critical scaling. Panel (d) illustrates the absence of a genuine reduced scaling for $\eta=-1$ (no physical critical point).
• Figure 5: Joule--Thomson inversion curves $T_i$ versus $P_i$ for the Mod(A)Max--AdS black hole with a cloud of strings, obtained from the condition $\mu_{JT}=0$. Panel (a) shows the effect of the charge parameter $Q$ at fixed $\gamma=0.5$, while panel (b) displays the dependence on the nonlinearity parameter $\gamma$ at fixed $Q=0.7$. In both panels we set $\eta=+1$ and $\alpha=0.1$ (the remaining parameters are the same as those adopted throughout this section).