We present a novel approach for probing the microstructure of a thermodynamic system that combines thermodynamic phase transitions with the Ruppeiner scalar curvature. Originally considered for van der Waals fluids and charged black holes [Phys. Rev. Lett. 123, 071103 (2019)], we extend and generalize our approach to higher-dimensional charged AdS black holes. Beginning with thermodynamic fluctuations, we construct the line element of the Ruppeiner geometry and obtain a universal formula for the scalar curvature $R$. We first review the thermodynamics of a van der Waals fluid and calculate the coexistence and spinodal curves. From this we are able to clearly display the phase diagram. Notwithstanding the invalidity of the equation of state in the coexistence phase regions, we find that the scalar curvature is always negative for the van der Waals fluid, indicating that attractive interactions dominate among the fluid microstructures. Along the coexistence curve, the scalar curvature $R$ decreases with temperature, and goes to negative infinity at a critical temperature. We then numerically study the critical phenomena associated with the scalar curvature, and find that the critical exponent is 2, and that $R(1\ensuremath{-}\stackrel{\texttildelow{}}{T}{)}^{2}{C}_{v}\ensuremath{\approx}1/8$, where $\stackrel{\texttildelow{}}{T}$ and ${C}_{v}$ are the respective reduced temperature and heat capacity. We next consider four-dimensional charged AdS black holes. Vanishing of the heat capacity at constant volume yields a divergent scalar curvature. In order to extract the corresponding information, we define a new scalar curvature that has behaviour similar to that of a van der Waals fluid. We analytically confirm that at the critical point of the small/large black hole phase transition, the scalar curvature has a critical exponent 2, and $R(1\ensuremath{-}\stackrel{\texttildelow{}}{T}{)}^{2}{C}_{v}=1/8$, the same as that of a van der Waals fluid. However we also find a significant distinction: the scalar curvature can be positive for the small charged AdS black hole, implying that repulsive interactions dominate among the black hole microstructures. We then generalize our study to higher-dimensional charged AdS black holes, and investigate the influence of the dimensionality on the black hole microstructures and the scalar curvature. Our novel approach provides a universal way for probing the microstructure of charged AdS black holes from a geometric construction.

Ruppeiner Geometry, Phase Transitions, and the Microstructure of Charged AdS Black Holes

The Joule-Thomson expansion process is studied for charged Gauss-Bonnet black holes in AdS space. First, in five-dimensional space-time, the isenthalpic curve in $T\ensuremath{-}P$ graph is obtained and the cooling-heating region is determined. Second, the explicit expression of Joule-Thomson coefficient is obtained from the basic formulas of enthalpy and temperature. Our methods can also be applied to van der Waals system as well as other black hole systems. And the inversion curve $\stackrel{\texttildelow{}}{T}(\stackrel{\texttildelow{}}{P})$ which separates the cooling region and heating region is obtained and investigated. Third, interesting dependence of the inversion curves on the charge ($Q$) and the Gauss-Bonnet parameter ($\ensuremath{\alpha}$) is revealed. In $\stackrel{\texttildelow{}}{T}\ensuremath{-}\stackrel{\texttildelow{}}{P}$ graph, the cooling region decreases with charge, but increases with the Gauss-Bonnet parameter. Fourth, by applying our methods, the Joule-Thomson expansion process for $\ensuremath{\alpha}=0$ case in four dimension is studied, where the Gauss-Bonnet AdS black hole degenerates into RN-AdS black hole. The inversion curves for van der Waals systems consist of two parts. One has positive slope, while the other has negative slope. However, for black hole systems, the slopes of the inversion curves are always positive, which seems to be a universal feature.

Joule-Thomson expansion of charged Gauss-Bonnet black holes in AdS space

Intriguing microstructures of five-dimensional neutral Gauss-Bonnet AdS black hole

We investigate the relations between the black hole shadow and charged AdS black hole critical behavior in the extended phase space. Using the thermo-shadow formalism built in Ref. 1, we reveal tha...

Thermal Image and Phase Transitions of Charged AdS Black Holes using Shadow Analysis

Effects of Lovelock gravity on the Joule-Thomson expansion are probed from various perspectives. The well-known Joule-Thomson coefficient is derived with both the explicit expression and intuitive image presented. Moreover, the inversion curves showing the relation between the inversion temperature and the inversion pressure are studied. It is shown that for given inversion pressure, the inversion temperature of the case $\alpha\neq0$ ($\alpha$ is the Lovelock parameter) is much lower than that of the case $\alpha=0$. And the inversion temperature tends to decrease with $\alpha$, in contrast to the effect of the electric charge. It is also shown that the ratio between the minimum inversion temperature and the critical temperature decreases with $\alpha$ for $\alpha\neq0$. Furthermore, the isenthalpic curves are investigated with rich physics revealed. The intersection point between the isenthalpic curve and the inversion curve is exactly the inversion point discriminating the heating process from cooling process. It is shown that both the inversion temperature and the inversion pressure for $\alpha\neq0$ are much lower for the same given mass of the black hole, showing the effect of Lovelock gravity. Last but not the least, we discuss the case of uncharged Lovelock AdS black holes with interesting feature found. It is shown that the Joule-Thomson coefficient is always positive, suggesting the expansion is always in the regime of cooling process. And no inversion temperature exists, in contrast to the case $Q\neq0$. Isenthalpic curves are also quite different since the temperature increases monotonically with the pressure when the mass is specified.

Effects of Lovelock gravity on the Joule–Thomson expansion

In this paper, we study Joule-Thomson effects for charged AdS black holes in $f(R)$ gravity. We obtain the inversion temperatures as well as inversion curves, and investigate similarities and differences between van der Waals fluids and charged AdS black holes in $f(R)$ gravity for this expansion. In addition, we determine the position of the inversion point versus different values of the mass $M$, the charge $Q$ and the parameter $b$ for such black hole. At this point, the Joule-Thomson coefficient $\mu$ vanishes, an import feature that we used to obtain the cooling-heating regions by scrutinizing the sign of the $\mu$ quantity.

Joule-Thomson Expansion of RN-AdS Black Holes in $f(R)$ gravity

In this work we study the phase transition of the charged-AdS black hole surrounded by quintessence via an alternative extended phase space defined by the charge square $Q^2$ and her conjugate $\Psi$, a quantity proportional to the inverse of horizon radius, while the cosmological constant is kept fixed. The equation of state is derived under the form $Q^2=Q^2(T,\Psi)$ and the critical behavior of such black hole analyzed. In addition, we explore the connection between the microscopic structure and Ruppeiner geothermodynamics. We also find that, at certain points of the phase space, the Ruppeiner curvature is characterized by the presence of singularities that are interpreted as phase transitions.

More Insight into Microscopic Properties of RN-AdS Black Hole Surrounded by Quintessence via an Alternative Extended Phase Space

Chaos in charged AdS black hole extended phase space

In this work, we study the phase transition of the charged-AdS black hole surrounded by quintessence via an alternative extended phase space defined by the charge square Q2 and her conjugate Ψ, a q...

More insight into microscopic properties of RN-AdS black hole surrounded by quintessence via an alternative extended phase space

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Joule-Thomson Expansion of RN-AdS Black Holes in $f(R)$ gravity

More Insight into Microscopic Properties of RN-AdS Black Hole Surrounded by Quintessence via an Alternative Extended Phase Space

Chaos in charged AdS black hole extended phase space

More insight into microscopic properties of RN-AdS black hole surrounded by quintessence via an alternative extended phase space