  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 $S_{BG}$ 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 \emph{area law}, so named
because, for a wide class of strongly quantum-entangled $d$-dimensional
systems, $S_{BG}$ is proportional to $\ln L$ if $d=1$, and to $L^{d-1}$ if
$d>1$, instead of being proportional to $L^d$ ($d \ge 1$). 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 \emph{not} to be
identified with the BG {\it additive} entropy but with appropriately
generalized {\it 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.

Statistical Mechanics