In contrast to alternative values, the quantum of area ∆A = 8πl 2 P does not follow from the usual statistical interpretation of black hole entropy; on the contrary, a statistical interpretation follows from it. This interpretation is based on the two concepts: nonadditivity of black hole entropy and Landau quantization. Using nonadditivity a microcanonical distribution for a black hole is found and it is shown that the statistical weight of black hole should be proportional to its area. By analogy with conventional Landau quantization, it is shown that quantization of black hole is nothing but the Landau quantization. The Landau levels of black hole and their degeneracy are found. The degree of degeneracy is equal to the number of ways to distribute a patch of area 8πl 2 P over the horizon. Taking into account these results, it is argued that the black hole entropy should be of the form S bh = 2π · ∆Γ, where the number of microstates is ∆Γ = A/8πl 2 P . The nature of the degrees of freedom responsible for black hole entropy is elucidated. The applications of the new interpretation are presented. The effect of noncommuting coordinates is discussed.

A diffusion equation for a black hole is derived from the BunsterCarlip equations. Its solution has the standard form of a Gaussian distribution. The second moment of the distribution determines the quantum of black hole area. The entropy of diffusion process is the same, apart from the logarithmic corrections, as the BekensteinHawking entropy.Bunster (Teitelboim) and Carlip showed [1] that the wave function of a black hole with the Arnowitt-Deser-Misner (ADM) mass M and area A evolves according to the Schrödinger-type equationswhere t is the lapse of asymptotic proper time at spatial infinity and Θ is the lapse of the hyperbolic angle at the horizon. Under Euclidean continuation Θ transforms to an angle variable. As a result, as pointed out in [1], A/8πG and Θ become conjugate exactly like M and t. This means that the area is the operator-valued quantity. It was shown in [2] that A/8πG can be 1

Proceeding in exactly the same way as in the derivation of the temperature of a dual CFT for the extremal black hole in the Kerr/CFT correspondence, it is found that the temperature of a chiral, dual CFT for the Schwarzschild black hole is T = 1/2π. Comparing Cardy's formula with the Bekenstein-Hawking entropy and using T , it is found that the central charge for the Schwarzschild black hole is of the form c = 12J in , where J in is the intrinsic angular momentum of the black hole, J in = A/8πG. It is shown that the central charge for any four-dimensional (4D) extremal black hole is of the same form. The possible universality of this form is briefly discussed.

A diffusion equation approach to black hole thermodynamics in Euclidean sector is proposed. A diffusion equation for a generic Kerr-Newman black hole in Euclidean sector is derived from the Bloch equation. Black hole thermodynamics is also derived and it is found, in particular, that the entropy of a Kerr-Newman black hole is the same, apart from the logarithmic corrections, as the Bekenstein-Hawking entropy of the black hole.

It is shown that the fast scrambling of information in a black hole can be viewed as Brownian motion of information in a fluid with negative viscosity (and negative temperature). It is argued that a non-local character of the fast scrambling is only an illusion; the stretched horizon with negative viscosity is an amplifying medium that mimics nonlocality and superluminal communication.