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

TL;DR

This paper reviews the relationship between statistical mechanics and thermodynamics, arguing that Boltzmann-Gibbs theory is sufficient but not necessary, and explores alternative thermostatistical theories for correlated systems.

Contribution

It demonstrates that nonadditive entropic frameworks can also satisfy thermodynamic principles, extending beyond traditional Boltzmann-Gibbs statistics.

Findings

Boltzmann-Gibbs statistics is sufficient but not necessary for thermodynamics.

Nonadditive entropic theories like q-statistics can also satisfy thermodynamic consistency.

Strongly correlated systems may fall outside the Boltzmann-Gibbs framework.

Abstract

We brief{}ly review the connection between statistical mechanics and thermodynamics. We show that, in order to satisfy thermodynamics and its Legendre transformation mathematical frame, the celebrated Boltzmann-Gibbs~(BG) statistical mechanics is suff{}icient but not necessary. Indeed, the $N\to\infty$ 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 (\emph{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.