Eur. Phys. J. Special Topics 223, 2161–2175 (2014)
https://doi.org/10.1140/epjst/e2014-02256-1

Review

Thermodynamics is more powerful than the role to it reserved by Boltzmann-Gibbs statistical mechanics

¹ Centro Brasileiro de Pesquisas Fisicas and National Institute of Science and Technology for Complex Systems, Rua Xavier Sigaud 150, 22290-180 Rio de Janeiro-RJ, Brazil
² Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA

^a e-mail: tsallis@cbpf.br
^b e-mail: cirto@cbpf.br

Received: 24 April 2014
Revised: 18 August 2014
Published online: 24 October 2014

Abstract

We briefly review the connection between statistical mechanics and thermodynamics. We show that, in order to satisfy thermo-dynamics and its Legendre transformation mathematical frame, the celebrated Boltzmann-Gibbs (BG) statistical mechanics is sufficient but not necessary. Indeed, the N →∞ limit of statistical mechanics is expected to be consistent with thermodynamics. For systems whose elements are generically independent or quasi-independent in the sense of the theory of probabilities, it is well known that the BG theory (based on the additive BG entropy) does satisfy this expectation. However, in complete analogy, other thermostatistical theories (e.g., q-statistics), based on nonadditive entropic functionals, also satisfy the very same expectation. We illustrate this standpoint with systems whose elements are strongly correlated in a specific manner, such that they escape the BG realm.