Area law: Canonical and microcanonical thermostatistics grounded on nonadditive entropies

Abstract The area law emerges in quantum 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: ( i.e. , if d = 1 and if ), instead of the expected standard scaling , where L characterizes the (dimensionless) linear size of the system which is focused on. Since, for such class of complex systems, the entropy S BG is nonextensive, the Legendre structure of thermodynamics is violated, in contrast with nonadditive entropic functionals such as S q with specific and S δ 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, and , in contrast with the BG theory which naturally yields .