Divergence Of Heat Capacity Research Articles

Fractional phase transitions of RN-AdS black holes at their Davies points* *Supported by the Shanxi Provincial Natural Science Foundation of China (201701D121002), the Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi (2021L386), and the Datong City Key Project of Research and Development of Industry of China (2018021).

In this study, we investigate the phase transitions of the RN-AdS black hole at its Davies points according to the generalized Ehrenfest classification of phase transition established based on fractional derivatives. Notably, Davies points label the positions at which the heat capacity diverges. According to the usual Ehrenfest classification, second-order phase transitions occur at these points. For the RN-AdS black hole, the Davies points can be classified into two types. The first type corresponds to extreme values of the temperature, and the second type corresponds to the infection point (namely the critical point) of temperature. Employing the generalized Ehrenfest classification, we determine that the orders of phase transition at the two types of Davies points are different, that is, we note an order of 3/2 for the first type and 4/3 for the second type. Thus, this finer-grained classification can discriminate between phase transitions that are expected to lie in the same category, providing new insights leading toward a better understanding of black hole thermodynamics.

Bardeen Black Holes in the Regularized 4D Einstein–Gauss–Bonnet Gravity

We obtain exact Bardeen black holes to the regularized 4D Einstein–Gauss–Bonnet (EGB) gravity minimally coupled with the nonlinear electrodynamics (NED). In turn, we analyze the horizon structure to determine the effect of GB parameter α on the minimum cutoff values of mass, M0, and magnetic monopole charge, g0, for the existence of a black hole horizon. We obtain an exact expression for thermodynamic quantities, namely, Hawking temperature T+, entropy S+, Helmholtz free energy F+, and specific heat C+ associated with the black hole horizon, and they show significant deviations from the 4D EGB case owing to NED. Interestingly, there exists a critical value of horizon radius, r+c, corresponding to the local maximum of Hawking temperature, at which heat capacity diverges, confirming the second-order phase transition. A discussion on the black holes of alternate regularized 4D EGB gravity belonging to the scalar-tensor theory is appended.

Thermodynamics and phase transition of rotating regular-de Sitter black holes

We analyze thermodynamic properties of the rotating regular black holes having mass ($M$), angular momentum ($a$), and a magnetic charge $(g)$, and encompass Kerr black hole ($g=0$). The mass $M$ has a minimum at the radius $r_+=r_+^{\star}$, where both the heat capacity and temperature vanish. The thermal phase transition is because of the divergence of heat capacity at a critical radius $r_{+}^C$ with stable (unstable) branches for $r_+<r_+^C$ ($>r_+^C$). We also generalize the rotating regular black holes in de Sitter (dS) background and analyzed its horizon structure to show that for each $g$, there are two critical values of the mass parameter $M_{\text{cr1}}$ and $M_{\text{cr2}}$ which correspond to the degenerate horizons. Thus, we have rotating regular-dS black holes with an additional cosmological horizon apart from the inner (Cauchy) and the outer (event) horizons. Next, we discuss the effective thermodynamic quantities of the rotating regular-dS black holes in the extended phase space where the cosmological constant ($\Lambda$) is considered as thermodynamic pressure. Combining the first laws at the two horizons, we calculate the heat capacity at constant pressure $C_P$, the volume expansion coefficient $\alpha$, and the isothermal compressibility $\kappa_T$. At a critical point, the specific heat at constant pressure, the volume expansion coefficient, and the isothermal compressibility of the regular-dS black holes exhibit an infinite peak suggesting a second-order phase transition.

Bardeen black hole surrounded by perfect fluid dark matter * *Support by the National Natural Science Foundation of China (11571342)

We derive an exact solution for a spherically symmetric Bardeen black hole surrounded by perfect fluid dark matter (PFDM). By treating the magnetic charge g and dark matter parameter as thermodynamic variables, we find that the first law of thermodynamics and the corresponding Smarr formula are satisfied. The thermodynamic stability of the black hole is also studied. The results show that there exists a critical radius where the heat capacity diverges, suggesting that the black hole is thermodynamically stable in the range . In addition, the critical radius increases with the magnetic charge g and decreases with the dark matter parameter . Applying the Newman-Janis algorithm, we generalize the spherically symmetric solution to the corresponding rotating black hole. With the metric at hand, the horizons and ergospheres are studied. It turns out that for a fixed dark matter parameter , in a certain range, with the increase of the rotation parameter a and magnetic charge g, the Cauchy horizon radius increases while the event horizon radius decreases. Finally, we investigate the energy extraction by the Penrose process in a rotating Bardeen black hole surrounded by PFDM.

Einstein[formula omitted]Gauss[formula omitted]Bonnet gravity with nonlinear electrodynamics

We obtain an exact spherically symmetric and magnetically charged black hole solution in 4D Einstein−Gauss−Bonnet gravity coupled with nonlinear electrodynamics with the Lagrangian LNED=−F∕(1+2∣βF∣) (F is the field strength invariant). The thermodynamics of the black hole is studied within our model. We calculate the Hawking temperature and the heat capacity of the black hole. The phase transitions occur in the point where the Hawking temperature possesses an extremum and the heat capacity diverges. It was shown that black holes are thermodynamically stable in some range of event horizon radii when the heat capacity and Hawking temperature are positive. The logarithmic correction to the Bekenstein−Hawking BH entropy is obtained which mimics a quantum correction. The dependence of the BH shadow radius on the model parameters is studied. We also investigate the particle emission rate of the BH. The stability of the BH is studied by analysing the quasinormal modes.

Phase transitions in Born-Infeld AdS black holes in D-dimensions

In this paper, we have investigated phase transitions in arbitrary spacetime dimensions for Born-Infeld AdS black holes. The phase transition points are characterised from the divergence of heat capacity of the black hole. Two well established techniques, namely, the Ehrenfest scheme and the Ruppeiner state space geometry approach are used to identify the order of phase transition the black hole undergoes. It is observed that the results obtained from these two methods agree with each other. Our analysis reveals that the phase transition is of second order. It is also observed from the variation of the heat capacity with entropy that the small unstable black hole phase becomes more and more stable with increase in the spacetime dimensions. We speculate that this dependence of the stability of the black hole on the spacetime dimension can put an upper limit to the dimension of spacetime from the physical condition of the improbability of the formation of a small stable black hole. We have also derived a Smarr relation in D-spacetime dimensions using scaling arguments and first law of black hole thermodynamics which includes the cosmological constant and the Born-Infeld parameters as thermodynamic variables.

Thermodynamics of rotating Bardeen black holes: Phase transitions and thermodynamics volume

We investigate the thermodynamical properties of the rotating Bardeen black holes characterized by mass $m$, the spin parameter $a$, and the magnetic charge $g$. We calculate exact expressions of the Hawking temperature, mass, entropy, and heat capacity. The black hole mass is minimum at radius ${r}_{+}^{E}$, where both the heat capacity and temperature vanish with a stable remnant. Also, there exists a critical radius ${r}_{+}^{C}$ of multiple orders, where the heat capacity diverges, suggesting that the black hole is thermodynamically stable in the range ${r}_{+}^{E}l{r}_{+}l{r}_{+}^{C}$. We also analyze the extended phase space thermodynamics of the rotating Bardeen--anti-de Sitter black holes. Treating the cosmological constant $\mathrm{\ensuremath{\Lambda}}$ and charge $g$ as thermodynamic variables, we derive the generalized first law in the extended phase space to study the critical phenomena of the black holes. The Ehrenfest scheme for the $P\ensuremath{-}V$ criticality of the rotating Bardeen black holes in anti-de Sitter spaces is examined. The Clausius-Clapeyron-Ehrenfest equations confirming the second-order phase transitions of the van der Waals fluid is no longer valid.

Black hole as a magnetic monopole within exponential nonlinear electrodynamics

We perform the gauge covariant quantization of the exponential model of nonlinear electrodynamics. Magnetically charged black holes, in the framework of our model are considered, and the regular black hole solution is obtained in general relativity. The asymptotic black hole solution at r→∞ is found. We calculate the magnetic mass of the black hole and the metric function which are expressed via the parameter β of the model and the magnetic charge. The thermodynamic properties and thermal stability of regular black holes are analysed. We calculate the Hawking temperature of black holes and their heat capacity at the constant magnetic charge. We find a point where the temperature changes the sign that corresponds to the first-order phase transition. It is shown that at critical point, where the heat capacity diverges, there is a phase transition of the second-order. We obtain the parameters of the model when the black hole is stable.

Thermodynamic Analysis of Cosmological Black Hole

The objective of this paper is to investigate the thermodynamic properties of McVittie universe. It turns out that the Einstein field equations are obtained by using Clausius equation near apparent horizon of McVittie universe. For the existence of horizons, different conditions in terms of McVittie parameters are discussed. Diverse cases of surface gravity are studied and conditions in terms of parameters for which surface gravity is positive, zero and negative are obtained and analyzed. It is shown that the generalized second law (GSL) holds near the apparent horizon if . Our results show that the heat capacity, CP, of the universe is always negative for matter dominated phase having P = 0. On the other hand, when , the heat capacity provided and negative otherwise. Furthermore, when , the temperature of the universe is minimum at while its heat capacity diverges.

Asymptotic Reissner-Nordström solution within nonlinear electrodynamics

A model of nonlinear electrodynamics coupled with the gravitational field is studied. We obtain the asymptotic black hole solutions at $r\rightarrow 0$ and $r\rightarrow \infty$. The asymptotic at $r\rightarrow 0$ is shown, and we find corrections to the Reissner-Nordstr\"om solution and Coulomb's law at $r\rightarrow\infty$. The mass of the black hole is evaluated having the electromagnetic origin. We investigate the thermodynamics of charged black holes and their thermal stability. The critical point corresponding to the second-order phase transition (where heat capacity diverges) is found. If the mass of the black hole is greater than the critical mass, the black hole becomes unstable.

Phase transition of charged Black Holes in Brans–Dicke theory through geometrical thermodynamics

In this paper, we take into account black hole solutions of Brans-Dicke-Maxwell theory and investigate their stability and phase transition points. We apply the concept of geometry in thermodynamics to obtain phase transition points and compare its results with those, calculated in canonical ensemble through heat capacity. We show that these black holes enjoy second order phase transitions. We also show that there is a lower bound for the horizon radius of physical charged black holes in Brans-Dicke theory which is originated from restrictions of positivity of temperature. In addition, we find that employing specific thermodynamical metric in the context of geometrical thermodynamics, yields divergencies for thermodynamical Ricci scalar in places of the phase transitions. It will be pointed out that due to characteristics behavior of thermodynamical Ricci scalar around its divergence points, one is able to distinguish the physical limitation point from the phase transitions. In addition, the free energy of these black holes will be obtained and its behavior will be investigated. It will be shown that the behavior of the free energy in the place where the heat capacity diverges, demonstrates second order phase transition characteristics.

Nonlinear electrodynamics and thermodynamic geometry of rotating dilaton black branes

We construct a new class of rotating dilaton solutions in the presence of logarithmic nonlinear electrodynamics. These solutions represent black branes with flat horizon and contain $$k=[(n-1)/2]$$ rotation parameters in n-dimensional spacetime where [x] is the integer part of x. We study the causal structure of the spacetime and calculate thermodynamic and conserved quantities and show that these quantities satisfy the first law of thermodynamics on the black brane horizon, $${ dM}={ TdS}+{{{\sum _{i=1}^{k}}}}\Omega _{i}d{J}_{i}+{ Ud}{Q}$$ . Then, we study geometrical approach towards thermodynamics by choosing an appropriate geometrical metric. We show that the singularity of the Ricci scalar coincides exactly with the phase transition points. We observe that our system encounters two types of phase transitions depending on the metric parameters. For the first one the heat capacity is zero and for the second one the heat capacity diverges. In the first kind of phase transition, the brane has a transition from an unstable non-physical to a stable physical state. In the second type of phase transition the brane moves from a stable to an unstable state. Finally, we comment on the dynamical stability of the obtained solutions under perturbations in four dimensions.

Pressure and Compressibility of Conformal Field Theories from the AdS/CFT Correspondence

The equation of state associated with N = 4 supersymmetric Yang–Mills in four dimensions, for S U ( N ) in the large N limit, is investigated using the AdS/CFT correspondence. An asymptotically AdS black-hole on the gravity side provides a thermal background for the Yang–Mills theory on the boundary in which the cosmological constant is equivalent to a volume. The thermodynamic variable conjugate to the cosmological constant is a pressure, and the P - V diagram of the quark-gluon plasma is studied. It is known that there is a critical point where the heat capacity diverges, and this is reflected in the isothermal compressibility. Critical exponents are derived and found to be mean field in the large N limit. The same analysis applied to three- and six-dimensional conformal field theories again yields mean field exponents associated with the compressibility at the critical point.

Incorporating critical divergence of isochoric heat capacity into the soft‐SAFT equation of state

In the critical region, widely used equations of state, including molecular‐based statistical associating fluid theory (SAFT) equations, fail to quantitatively describe the critical anomalies affected by the diverging fluctuations of density due to their classical mean‐field approximation. Description of some properties in the critical region, in particular vapor‐liquid coexistence and isothermal compressibility, can be improved by fitting (multi) parameters of an equation of state to the properties obtained in accurate computer simulations. However, some anomalies in the critical region, which are caused exclusively by critical fluctuations, cannot be readily generated by this procedure. An example of a critical anomaly, completely absent in the mean‐field approximation, is the divergence of the isochoric heat capacity. A simplified methodology for modifying the original soft‐SAFT equation of state to take into account these anomalies is explored. Possible procedures for further improvement are mentioned. © 2015 American Institute of Chemical Engineers AIChE J, 61: 3073–3080, 2015

Bose condensation and branes

When the cosmological constant is considered to be a thermodynamical variable in black hole thermodynamics, analogous to a pressure, its conjugate variable can be thought of as a thermodynamic volume for the black hole. In the AdS/CFT correspondence this interpretation cannot be applied to the CFT on the boundary but, from the point of view of the boundary $SU(N)$ gauge theory, varying the cosmological constant in the bulk is equivalent to varying the number of colors in the gauge theory. This interpretation is examined in the case of $AdS_5\times S^5$, for ${\cal N}=4$ SUSY Yang-Mills at large $N$, and the variable thermodynamically conjugate to $N$, a chemical potential for color, is determined. It is shown that the chemical potential in the high temperature phase of the Yang-Mills theory is negative and decreases as temperature increases, as expected. For spherical black holes in the bulk the chemical potential approaches zero as the temperature is lowered below the Hawking-Page temperature and changes sign at a temperature that is within one part in a thousand of the temperature at which the heat capacity diverges.

Critical anomalies of alkaline fading of phenolphthalein in the critical solution of 2-butoxyethanol + water

We have used three-wavelength UV-spectrophotometry to study the reaction of the alkaline fading of phenolphthalein in the critical solution of 2-butoxyethanol + water. It was found that when the temperature was far away from the critical point, the values of the natural logarithm of the rate constant k and the natural logarithm of the chemical equilibrium K determined in our experiments had good linear relationships with the reciprocal of temperature, which served as the backgrounds and were used for correcting k and K in the critical region. The critical slowing down of the reaction and the critical anomaly of the chemical equilibrium were detected near the critical point. The value of the critical exponent characterizing the slowing down effect of the reaction rate was obtained to be 0.156, which was close to the value 0.11 associated with the heat capacity divergence and agreed with the theoretical prediction. The experimental result also confirmed the theoretical prediction of 0.11 for the critical exponent characterizing the weak divergence of the singularity of the chemical equilibrium.

Heat capacity singularity of binary liquid mixtures at the liquid-liquid critical point

The critical anomaly of the isobaric molar heat capacity for the liquid-liquid phase transition in binary nonionic mixtures is explained through a theory based on the general assumption that their partition function can be exactly mapped into that of the Ising three-dimensional model. Under this approximation, it is found that the heat capacity singularity is directly linked to molar excess enthalpy. In order to check this prediction and complete the available data for such systems, isobaric molar heat capacity and molar excess enthalpy near the liquid-liquid critical point were experimentally determined for a large set of binary liquid mixtures. Agreement between theory and experimental results-both from literature and from present work-is good for most cases. This fact opens a way for explaining and predicting the heat capacity divergence at the liquid-liquid critical point through basically the same microscopic arguments as for molar excess enthalpy, widely used in the frame of solution thermodynamics.

Geometrothermodynamics of five dimensional black holes in Einstein–Gauss–Bonnet theory

We investigate the thermodynamic properties of 5D static and spherically symmetric black holes in (i) Einstein-Maxwell-Gauss-Bonnet theory, (ii) Einstein-Maxwell-Gauss-Bonnet theory with negative cosmological constant, and in (iii) Einstein-Yang-Mills-Gauss-Bonnet theory. To formulate the thermodynamics of these black holes we use the Bekenstein-Hawking entropy relation and, alternatively, a modified entropy formula which follows from the first law of thermodynamics of black holes. The results of both approaches are not equivalent. Using the formalism of geometrothermodynamics, we introduce in the manifold of equilibrium states a Legendre invariant metric for each black hole and for each thermodynamic approach, and show that the thermodynamic curvature diverges at those points where the temperature vanishes and the heat capacity diverges.

Letter to the Editor: Fifty Years of Breakthrough Discoveries in Fluid Criticality

Fifty years ago two scientists, who celebrate their 80th birthdays in 2011, Alexander V. Voronel and Johannes V. Sengers performed breakthrough experiments that challenged the commonly accepted views on critical phenomena in fluids. Voronel discovered that the isochoric heat capacity of argon becomes infinite at the vapor-liquid critical point. Almost simultaneously, Sengers observed a similar anomaly for the thermal conductivity of near-critical carbon dioxide. The existence of these singularities was later proved to be universal for all fluids. These experiments had a profound effect on the development of the modern (scaling) theory of phase transitions, which is based on the diverging fluctuations of the order parameter. In particular, the discovery of the heat-capacity divergence at the critical point was a keystone for the formulation of static scaling theory, while the discovery of the divergence of the thermal conductivity played an important role in the formulation of dynamic scaling and mode-coupling theory. Moreover, owing to the discoveries made by Voronel and Sengers 50 years ago, critical phenomena in fluids have become an integral part of contemporary condensed-matter physics.

New type of phase transition in Reissner Nordström–AdS black hole and its thermodynamic geometry

The phase transition of a RN–AdS black hole is studied in details using Ehrenfest's equations. There is no discontinuity in entropy which signals a lack of any first order phase transition. We then show that although Ehrenfest's first equation is satisfied, the second is not, so that a true second order phase transition is also ruled out. However this deviation from the second Ehrenfest's equation, for a certain range of the black hole charge, indicates the existence of a new glassy type transition. We finally study the thermodynamic behaviour using state space geometry and find that the scalar curvature diverges exactly at those points where the heat capacity diverges.