Some statistical approaches in the apparent horizon entropy and the generalized second law of thermodynamics

The well known connection between black holes and thermodynamics, as well as their basic statistical mechanics, has been explored during the last decades since the published papers by Hawking, Jacobson and Unruh. In this work we have investigated the effects of three nongaussian entropies which are the modified Renyi entropy (MRE), Sharma-Mittal entropy (SME) and the dual Kaniadakis entropy (DKE) in the investigation of the generalized second law of thermodynamics, an extension of second law for black holes. Recently, it was analyzed that a total entropy is the sum of the entropy enclosed by the apparent horizon plus the entropy of the horizon itself when the apparent horizon is described by the Barrow entropy. It is assumed that the universe is filled with the matter and dark energy fluids. Here, as we said just above, the apparent horizon is described by the MRE and SME entropies, and then by the DKE proposal. We have established conditions where the second law of thermodynamics can or cannot be obeyed in the MRE, the SME as well as in the DKE just as it did in Barrows entropy.

Barrow black hole corrected-entropy model and Tsallis nonextensivity

The quantum scenario concerning Hawking radiation, gives us a precious clue that a black hole has its temperature directly connected to its area gravity and that its entropy is proportional to the horizon area. These results have shown that there exist a deep association between thermodynamics and gravity. The recently introduced Barrow formulation of back holes entropy, influenced by the spacetime geometry, shows the quantum fluctuations effects through Barrow exponent, $Delta$, where $Delta=0$ represents the usual spacetime and its maximum value, $Delta=1$, characterizes a fractal spacetime. The quantum fluctuations are responsible for such fractality. Loop quantum gravity approach provided the logarithmic corrections to the entropy. This correction arises from quantum and thermal equilibrium fluctuations. In this paper we have analyzed the nonextensive thermodynamical effects of the quantum fluctuations upon the geometry of a Barrow black hole. We discussed the Tsallis formulation of this logarithmically corrected Barrow entropy to construct the equipartition law. Besides, we obtained a master equation that provides the equipartition law for any value of the Tsallis $q$-parameter and we analyzed several different scenarios. After that, the heat capacity were calculated and the thermal stability analysis was carried out as a function of the main parameters, namely, one of the so-called pre-factors, $q$ and $Delta$.

On the nature of Renyi modified entropy and the Incomplete statistics approach in black holes thermodynamics

In this work we have investigated the effects of the two highlighted nongaussian entropies which are the modified Renyi entropy and the so-called Incomplete statistics in the analysis of the thermodynamics of black holes (BHs). We have obtained the equipartition theorems and after that we obtained the heat capacities for both approaches. Depending on the values of both $lambda-$parameter and $M$, the BH mass, relative to a modified Renyi entropy and, depending on the values of the $q-$parameter and $M$, the Incomplete entropy can determine if the BH has an unstable thermal equilibrium or not for each model.

Barrows black hole entropy and the equipartition theorem

The Barrow entropy appears from the fact that the black hole surface can be modified due to quantum gravitational outcome. The measure of this perturbation is given by a new exponent $Delta$. In this letter we have shown that, from the standard mathematical form of the equipartition theorem, we can relate it with Barrow entropy. From this equivalence, we have calculated precisely the value of the exponent for the equipartition law. After that, we tested the thermodynamical coherence of the system by calculating the heat capacity which established an interval of the possible thermodynamical coherent values of Barrow entropic exponent and corroborated our first result.

On the equipartition theorem and black holes nongaussian entropies

In this Letter we have shown that, from the standard thermodynamic functions, the mathematical form of an equipartition theorem may be related to the algebraic expression of a particular entropy initially chosen to describe the black hole thermodynamics. Namely, we have different equipartition expressions for distinct statistics. To this end, four different mathematical expressions for the entropy have been selected to demonstrate our objective. Furthermore, a possible phase transition is observed in the heat capacity behavior of the Tsallis and Cirto entropy model.

Black holes quasinormal modes, Loop Quantum Gravity Immirzi parameter and nonextensive statistics

It is argued that, using the black hole area entropy law together with the Boltzmann-Gibbs statistical mechanics and the quasinormal modes of the black holes, it is possible to determine univocally the lowest possible value for the spin $j$ in the context of the Loop Quantum Gravity theory which is $j_{min}=1$. Consequently, the value of Immirzi parameter is given by $gamma = ln 3/(2pisqrt{2})$. In this paper, we have shown that if we use Tsallis microcanonical entropy rather than Boltzmann-Gibbs framework then the minimum value of the label $j$ depends on the nonextensive $q$-parameter and may have values other than $j_{min}=1$.

The unphysical character of dark energy fluids

It is well known that, in the context of general relativity, an unknown kind of matter that must violate the strong energy condition is required to explain the current accelerated phase of expansion of the Universe. This unknown component is called dark energy and is characterized by an equation of state parameter $w=p/rho<-1/3$. Thermodynamic stability requires that $3w-dln |w|/dln age0$ and positiveness of entropy that $wge-1$. In this paper we proof that we cannot obtain a differentiable function $w(a)$ to represent the dark energy that satisfies these conditions trough the entire history of the Universe.

Tsallis and Kaniadakis statistics from the viewpoint of entropic gravity formalism

It has been shown in the literature that effective gravitational constants, which are derived from Verlindes formalism, can be used to introduce the Tsallis and Kaniadakis statistics. This method provides a simple alternative to the usual procedure normally used in these non-Gaussian statistics. We have applied our formalism in the Jeans mass criterion of stability and in the free fall time collapsing of a self-gravitating system where new results are obtained. A possible connection between our formalism and deviations of Newtons law of gravitation in a submillimeter range is made.

Jeans instability criterion from the viewpoint of non-gaussian statistics

In this Letter we have derived the Jeans length in the context of the Kaniadakis statistics. We have compared this result with the Jeans length obtained in the non-extensive Tsallis statistics and discussed the main differences between these two models. We have also obtained the kappa-sound velocity. Finally, we have applied the results obtained here to analyze an astrophysical system.

Quantum Gravitational Correction and MOND Theory in the Holographic Equipartition Scenario

In this paper, by using Verlindes formalism and a modified Padmanabhans prescription, we have obtained the lowest order quantum correction to the gravitational acceleration and MOND-type theory by considering a nonzero difference between the number of bits of the holographic screen and number of bits of the holographic screen that satisfy the equipartition theorem. We will also carry out a phase transition and critical phenomena analysis in MOND-type theory where critical exponents are obtained.