A Comprehensive Review of Geometrical Thermodynamics: From Fluctuations to Black Holes

In this paper, we consider linearly charged dilatonic black holes and study their thermodynamical behavior in the context of phase transitions and thermodynamic geometry. We show that, depending on the values of the parameters, these black holes can undergo two types of phase transition. We also find that there are three critical behaviors near the critical points for these black holes: nonphysical unstable to physical stable, large to small, and small to large black-hole phase transitions. Next, we employ a thermodynamical metric to study the thermodynamical geometry of these black holes. We show that the characteristic behavior of the Ricci scalar of this metric enables one to recognize the type of phase transition and critical behavior of the black holes near phase-transition points. Finally, we will extend thermodynamical space by considering the dilaton parameter as an extensive parameter. With this consideration, we will show that the Weinhold, Ruppeiner, and Quevedo metrics prov …

The application of information geometric ideas to statistical mechanics using a metric on the space of states, as pioneered by Ruppeiner and Weinhold, has proved to be a useful alternative approach to characterizing phase transitions. Some puzzling anomalies become apparent, however, when these methods are applied to the study of black hole thermodynamics. A possible resolution was suggested by Quevedo et al who emphasized the importance of Legendre invariance in thermodynamic metrics. They found physically consistent results for various black holes when using a Legendre invariant metric, which agreed with a direct determination of the properties of phase transitions from the specific heat. Recently, information geometric methods have been employed by Wei et al to study the Kehagias–Sfetsos (KS) black hole in Hořava–Lifshitz gravity. The formalism suggests that a coupling parameter in this theory plays a role analogous to the charge in Reissner–Nordström black holes or angular momentum …

We consider three-dimensional BTZ black holes with three models of nonlinear electrodynamics as source. Calculating heat capacity, we study the stability and phase transitions of these black holes. We show that Maxwell, logarithmic, and exponential theories yield only type one phase transition which is related to the root(s) of heat capacity, whereas, for correction form of nonlinear electrodynamics, heat capacity contains two roots and one divergence point. Next, we use geometrical approach for studying classical thermodynamical behavior of the system. We show that Weinhold and Ruppeiner metrics fail to provide fruitful results and the consequences of the Quevedo approach are not completely matched to the heat capacity results. Then, we employ a new metric for solving this problem. We show that this approach is successful and all divergencies of its Ricci scalar and phase transition points coincide. We also show that there is no phase transition for uncharged BTZ black holes.

We study the thermodynamics and thermodynamic geometry of a five-dimensional Reissner-Nordstr\"om-AdS black hole in the extended phase space by treating the cosmological constant as being related to the number of colors in the boundary gauge theory and its conjugate quantity as the associated chemical potential. We find that the contribution of the charge of the black hole to the chemical potential is always positive, and the existence of charge makes the chemical potential become positive more easily. We calculate the scalar curvatures of the thermodynamical Weinhold metric, Ruppeiner metric, and Quevedo metric, respectively, in the fixed ${N}^{2}$ case and the fixed $q$ case. We find that in the fixed ${N}^{2}$ case, the divergence of the scalar curvature is related to the divergence of the specific heat with fixed electric potential in the Weinhold metric and Ruppeiner metric, and the divergence of the scalar curvature in the Quevedo metric corresponds to the divergence of the speci …

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 the 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 originates from restrictions of positivity of temperature. In addition, we find that employing a specific thermodynamical metric in the context of geometrical thermodynamics yields divergencies for the thermodynamical Ricci scalar in places of the phase transitions. It will be pointed out that due to the characteristic behavior of the thermodynamical Ricci scalar around its divergence points, one is able to distinguish the physical limitation point from the phase transitio …