Area law: Canonical and microcanonical thermostatistics grounded on nonadditive entropies

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: S₁₆ (L) L^{d-1-1}d-1\; (L ;\, d 1) (i. e. , S₁₆ (L) L if d = 1 and S₁₆ (L) L^d-1 if d>1), instead of the expected standard scaling S₁₆ (L) L^d, 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 1 which yield extensive entropies, being thus consistent with classical thermodynamics. We discuss here the corresponding canonical and microcanonical thermostatistics and argue that, generically, q₌₈₂ₑ₎₂₀₍₎₍₈₂₀₋ ₂₀₍₎₍₈₂₀₋>1, in contrast with the BG theory which naturally yields q₌₈₂ₑ₎₂₀₍₎₍₈₂₀₋= q₂₀₍₎₍₈₂₀₋= ₌₈₂ₑ₎₂₀₍₎₍₈₂₀₋= ₂₀₍₎₍₈₂₀₋=1.