The Boltzmann–Gibbs–von Neumann entropy of a large part (of linear size $L$) of some (much larger) $d$-dimensional quantum systems follows the so-called area law (as for black holes), i.e., it is proportional to $L^{d-1}$. Here we show, for $d=1,2$, that the (nonadditive) entropy $S_q$ satisfies, for a special value of $q\ne 1$, the classical thermodynamical prescription for the entropy to be extensive, i.e., $S_q\propto L^d$. Therefore, we reconcile with classical thermodynamics the area law widespread in quantum systems. Recently, a similar behavior was exhibited in mathematical models with scale-invariant correlations [C. Tsallis, M. Gell-Mann, and Y. Sato, Proc. Natl. Acad. Sci. U.S.A.102 15377 (2005)]. Finally, we find that the system critical features are marked by a maximum of the special entropic index $q$.

• Received 16 March 2008

DOI:https://doi.org/10.1103/PhysRevE.78.021102