Thermo-Statistical description of the Hamiltonian non extensive systems: The selfsimilarity scaling laws

The foundations for a thermo-statistical description of the called non extensive Hamiltonian systems are reconsidered. The relevance of the parametric resonance as a fundamental mechanism of the Hamiltonian chaoticity in those systems with bound motions in the configurational space is discussed. The universality of this mechanism suggests us the possibility of performing a thermo-statistical description with microcanonical basis in the context of the long-range interacting Hamiltonian systems. The concept of selfsimilarity is proposed as an appropriate generalization of the well-known extensive conditions exhibited by the traditional systems, which is used to justify a given generalized thermodynamic formalism starting from the consideration of the microcanonical ensemble, i.e. the nonextensive Statistics of Tsallis. These ideas are illustrated by considering a recent proposed astrophysical model based on the quasi-ergodic character of the microscopic dynamics of these paradigmatic examples of real long-range interacting systems.