Area law: Canonical and microcanonical thermostatistics grounded on nonadditive entropies

Abstract The area law emerges in quantum (d + 1)-dimensional systems such as zero-temperature critical
phenomena as well as black-holes (and related cosmological models). For wide classes of systems,
it can be expressed as the following anomalous scaling of the Boltzmann-Gibbs-von Neumann entropy
SBG(L) ∝ Ld−1−1
d−1 (L → ∞; d ≥ 1) ( i.e., SBG(L) ∝ lnL if d = and SBG(L) ∝ Ld−1 if
d > 1), instead of the expected standard scaling SBG(L) ∝ Ld, where L characterizes the (dimensionless)
linear size of the system which is focused on. Since, for such class of complex systems,
the entropy SBG is nonextensive, the Legendre structure of thermodynamics is violated, in contrast
with nonadditive entropic functionals such as Sq with specific q < and Sδ with specific δ > 1
which yield extensive entropies, consistently with thermodynamics. We discuss
here the corresponding canonical and microcanonical thermostatistics and argue that, generically,
qmicrocanonical < qcanonical < and δmicrocanonical > δcanonical > 1, in contrast with the BG theory
which naturally yields qmicrocanonical = qcanonical = δmicrocanonical = δcanonical = 1.