Microcanonical foundation of nonextensivity and generalized thermostatistics based on the fractality of the phase space

TL;DR

This paper develops a unified microcanonical framework for statistical mechanics that naturally derives Boltzmann-Gibbs and Tsallis thermostatistics, linking the entropic index to phase space fractality.

Contribution

It introduces a generalized microcanonical approach that connects phase space fractality with nonextensive thermostatistics, providing a geometric interpretation of the entropic index q.

Findings

Derives Boltzmann-Gibbs statistics for smooth phase spaces.

Obtains Tsallis statistics for fractal phase spaces.

Links the entropic index q to the fractal dimension of phase space.

Abstract

We develop a generalized theory of (meta)equilibrium statistical mechanics in the thermodynamic limit valid for both smooth and fractal phase spaces. In the former case, our approach leads naturally to Boltzmann-Gibbs standard thermostatistics while, in the latter, Tsallis thermostatistics is straightforwardly obtained as the most appropriate formalism. We first focus on the microcanonical ensemble stressing the importance of the limit $t \to \infty$ on the form of the microcanonical measure. Interestingly, this approach leads to interpret the entropic index $q$ as the box-counting dimension of the (microcanonical) phase space when fractality is considered.