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

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→∞ 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.