Black hole thermodynamical entropy

As early as 1902, Gibbs pointed out that systems whose partition function diverges, e.g. gravitation, lie outside the validity of the Boltzmann-Gibbs (BG) theory. Consistently, since the pioneering Bekenstein-Hawking results, physically meaningful evidence (e.g., the holographic principle) has accumulated that the BG entropy SBG of a (3+1) black hole is proportional to its area L-2 (L being a characteristic linear length), and not to its volume L-3. Similarly it exists the area law, so named because, for a wide class of strongly quantum-entangled d-dimensional systems, SBG is proportional to lnL if d = 1, and to Ld-1 if d > 1, instead of being proportional to L-d (d >= 1). These results violate the extensivity of the thermodynamical entropy of a d-dimensional system. This thermodynamical inconsistency disappears if we realize that the thermodynamical entropy of such nonstandard systems is not to be identified with the BG additive entropy but with appropriately generalized nonadditive entropies. Indeed, the celebrated usefulness of the BG entropy is founded on hypothesis such as relatively weak probabilistic correlations (and their connections to ergodicity, which by no means can be assumed as a general rule of nature). Here we introduce a generalized entropy which, for the Schwarzschild black hole and the area law, can solve the thermodynamic puzzle.