The inner structure of realistic materials make them exhibit momentum relaxation. In this paper we study the holographic version of the Joule-Thomson effect on AdS black holes in which translational invariance is broken by two methods: First by considering planar black holes in general relativity supported by axion scalar fields with a linear dependence on the horizon coordinates and secondly by considering black holes in massive gravity models in which momentum relaxation is obtained by breaking the bulk diffeomorphism invariance of the theory. In contrast with black holes studied so far, for both theories it is possible to obtain inversion curves with two branches reproducing the behavior of Van der Wall fluids. Moreover in the specific case of the massive gravity model we show that black holes can heat up when crossing the inversion curve.

Recently a number of papers have claimed that the horizon area -and thus the entropy -of near extremal black holes in anti-de Sitter spacetimes can be reduced by dropping particles into them. In this note we point out that this is a consequence of an underlying assumption that the energy of an infalling particle changes only the internal energy of the black hole, whereas a more physical assumption would be that it changes the enthalpy (mass). In fact, under the latter choice, the second law of extended black hole thermodynamics is no longer violated.

We study the extended thermodynamical properties of the charged black hole in Horndeski model with the k-essence sector. Then we define a holographic heat engine via the black hole.We compute the engine efficiency in the large temperature limit and compare the results with the exact ones. With the given specified parameters in the rectangular engine, the higher order coupling suppresses the engine efficiencies.

The Smarr relation plays an important role in black hole thermodynamics. It is often claimed that the Smarr relation can be written down simply by observing the scaling behavior of the various thermodynamical quantities. We point out that this is not necessarily so in the presence of dimensionful coupling constants, and discuss the issues involving the identification of thermodynamical variables. I. SMARR RELATION AND THE FIRST LAWThe fact that black holes behave like a thermodynamical system has dramatically changed our understanding of black holes ever since its conception in 1973 [1]. For an asymptotically flat Kerr-Newman black hole, the first law of black hole mechanics takes the formwhere M denotes the ADM mass of the black hole, S its Bekenstein-Hawking entropy, T its Hawking temperature, Q its electrical charge and J its angular momentum. The first law thus relates the various differential quantities. In some applications, one would like to work directly with the black hole parameters instead of their differentials. Fortunately, there is the Smarr relation [2]:where Φ denotes the electrical potential, while Ω denotes the angular velocity of the black hole. Smarr relations such as this have been widely studied in the literature, beyond the Kerr-Newman family. A good rule of thumb for writing down the Smarr relation for a given black hole is to look at the scaling (i.e. the dimensions) of the various thermodynamical quantities. See, e.g., Sec.2 of [3]. For example, in 4-dimensions, and in the units = k B = c = 1, we have M, Q ∝ L, and J, S ∝ L 2 , where L is a length scale. Due to Euler's theorem of quasi-homogeneous function (see below), we can simply write down M = M (S, Q, J) asFrom the first law (and the chain rule), one could identify the various partial derivatives and arrive at the Smarr relation, Eq.(2). Similarly, in the extended black hole

In this study, we apply two methods to consider the variation of massive black holes in both normal and extended thermodynamic phase spaces. The first method considers a charged particle being absorbed by the black hole, whereas the second considers a shell of dust falling into it. With the former method, the first and second laws of thermodynamics are always satisfied in the normal phase space; however, in the extended phase space, the first law is satisfied but the validity of the second law of thermodynamics depends upon the model parameters. With the latter method, both laws are valid. We argue that the former method's violation of the second law of thermodynamics may be attributable to the assumption that the change of internal energy of the black hole is equal to the energy of the particle. Finally, we demonstrate that the event horizon always ensures the validity of weak cosmic censorship in both phase spaces; this means that the violation of the second law of thermodynamics, arising under the aforementioned assumption, does not affect the weak cosmic censorship conjecture. This further supports our argument that the assumption in the first method is responsible for the violation and requires deeper treatment.