Constantino Tsallis, Leonardo J. L. Cirto
As early as 1902, Gibbs pointed out that systems with long-range
interactions, like gravitation, lie outside the validity of standard
statistical mechanics. Nevertheless, the entropy of a black hole has been
repeatedly calculated within Boltzmann-Gibbs concepts. Since the pioneering
Bekenstein-Hawking results, it has become common to state that the black-hole
"entropy" is proportional to its area. Similarly it exists the {\it area law},
so named because the "entropy" of a wide class of $d$-dimensional quantum
systems is proportional to the $d$-dimensional area $A_d = L^{d-1}$ ($d>1$; $L$
is a characteristic length), instead of the $d$-dimensional volume $V_d =
A_d^{d/(d-1)} = L^d$. These results violate the extensivity of the
thermodynamical entropy. This inconsistency disappears if we realize that the
entropies of such nonstandard systems must {\it not} be associated with the
additive expression $S_{BG}=k_B\ln{W}$ but with appropriate nonadditive
generalizations. Here we introduce a generalized form of entropy which solves
the puzzle for the black hole and the area law.
View original:
http://arxiv.org/abs/1202.2154
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