Area law: Canonical and microcanonical thermostatistics grounded on nonadditive entropies

Centro Brasileiro de Pesquisas Físicas and National Institute of Science and Technology for Complex Systems Rua Dr. Xavier Sigaud 150, 22290-180, Rio de Janeiro, Brazil, Santa Fe Institute - 1399 Hyde Park Road, 87501, Santa Fe, NM, USA and Complexity Science Hub - Metternichgasse 8, 1030 Vienna, Austria

Received: 27 January 2026
Accepted: 3 March 2026

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 .