Area law: Canonical and microcanonical thermostatistics grounded on nonadditive entropies

The area law emerges in quantum $(d+1)\text{-dimensional}$ 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: $S_{BG}(L) \propto \frac {L^{d-1}-1}{d-1}\;(L\to \infty ;\,d \ge1)$ (i.e., $S_{BG}(L) \propto \ln L$ if d = 1 and $S_{BG}(L)\propto L^{d-1}$ if $d>1$ ), instead of the expected standard scaling $S_{BG}(L) \propto L^{d}$ , 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 $q<1$ and Sδ with specific δ> 1 which yield extensive entropies, being thus consistent with classical thermodynamics. We discuss here the corresponding canonical and microcanonical thermostatistics and argue that, generically, $q_{microcanonical}< q_{canonical}<1$ and $\delta _{microcanonical}>\delta _{canonical}>1$ , in contrast with the BG theory which naturally yields $q_{microcanonical}= q_{canonical}=\delta _{microcanonical}= \delta _{canonical}=1$ .