We present an analytical study of chaos in a charged black hole in the extended phase space in the context of the Poincare -Melnikov theory. Along with some background on dynamical systems, we compute the relevant Melnikov function and find its zeros. Then we analyse these zeros either to identify the temporal chaos in the spinodal region, or to observe spatial chaos in the small/large black hole equilibrium configuration. As a byproduct, we derive a constraint on the Black hole' charge required to produce chaotic behaviour. To the best of our knowledge, this is the first endeavour to understand the correlation between chaos and phase picture in black holes.

In this work we use the quasinormal frequencies of a massless scalar perturbation to probe the phase transition of the high dimension charged AdS black hole. The signature of the critical behavior of this black hole solution is detected in the isobaric as well as in isothermal process. This paper is a natural generalization of Liu et al. (JHEP 1409(JHEP :179, 2014 to higher dimensional spacetime. More precisely our study shows a clear signal for any dimension d in the isobaric process. As to the isothermal case, we find that this signature can be affected by other parameters like the pressure and the horizon radius. We conclude that the quasinormal modes can be an efficient tool to investigate the first-order phase transition, but fail to disclose the signature of the second-order phase transition.

In this paper, we study Joule-Thomson expansion 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 µ vanishes, an import feature that we used to obtain the cooling-heating regions by scrutinizing the sign of the µ quantity.Nowadays, an f (R) gravity is one of important class of the modified Einstein's gravity. In general, It is built through adding higher powers of the scalar curvature R, the Riemann and Ricci tensors, or their derivatives to the lagrangian description [1][2][3][4][5]. This kind of gravity mimics successfully the history of universe, especially the current cosmic acceleration, the inflation and structure formation in the early Universe [2,3], various extensions of f(R) gravity theory has been elaborated ranging from three dimensional [6] and asymptotically Lifshitz black hole solutions [7] to F(R) gravity's rainbow model [8].Recently, a special attention has been devoted to the study of the black holes phase transition particularly after the introduction of the notion of the extended phase space via the identification of the cosmological constant with the pressure and its conjugate quantity with thermodynamic volume [9]. In this context, the black hole behaves like Van der Waals fluid [10-12] leading to a remarkable correspondence between the thermal physics of black holes and simple substances [13]. Numerous extensions of these works have been elaborated for rotating and hairy black hole [14,15], high curvature theories of gravity and M-theory [16][17][18][19]. More exotic results as holographic heat engine [20] as well as other technics ranging from the behaviour of the quasi-normal modes [21,22] to AdS/CFT tools [23,24] and chaos structure [25] have consolidated the similarity with the Van der Waals fluid.Meanwhile, a considerable effort has been dedicated to explore the thermodynamics of the AdS black hole in the R + f (R) gravity background with a constant curvature [26] where the essential of thermodynamical quantities like the entropy, heat capacity and the Helmholtz free energy are calculated, then the extended phase space and the critical Van der Waals-like behavior introduced in [26,27] as well as the canonical ensembles in [28]. Here we would like to go further in the thermodynamical investigation and study the Joule-Thomson expansion for the charged-AdS black hole configuration in the f (R) gravity background.More recently, the authors of [29] have investigated the JT expansion for AdS charged black holes with the aim to confront the resulting features with those of Van der Waals fluids. The extension to rotating-AdS black hole [30] and the charged black hole solution in the presence of the ...

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 Ψ, 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, Ψ) 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.

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