Gutenberg-Richter, Omori and Cumulative Benioff strain patterns in terms of non extensive statistical physics and Beck-Cohen Superstatistics.

     The earthquake generation process is a complex phenomenon, manifested in the nonlinear dynamics and in the wide range of spatial and temporal scales that are incorporated in the process. Despite the complexity of the earthquake generation process and our limited knowledge on the physical processes that lead to the initiation and propagation of a seismic rupture giving rise to earthquakes, the collective properties of many earthquakes present patterns that seem universally valid. The most prominent is scale-invariance, which is manifested in the size of faults, the frequency of earthquake sizes and the spatial and temporal scales of seismicity.                                                                                                                                                                The frequency magnitude distribution exhibits a decay that is commonly expressed with the well-known Gutenberg-Richter (G-R) law. The aftershock production rate following a main event generally decays as a power-law with time according to the modified Omori formula. Scale-invariance and (multi)fractality are also manifested in the temporal evolution of seismicity and the distribution of earthquake epicentres. The organization patterns that earthquakes and faults exhibit have motivated the statistical physics approach to earthquake occurrence. Based on statistical physics and the entropy principle, a unified framework that produces the collective properties of earthquakes and faults from the specification of their microscopic elements and their interactions, has recently been introduced. This framework, called nonextensive statistical mechanics (NESM) was introduced as a generalization of classic statistical mechanics due to Boltzmann and Gibbs (BG), to describe the macroscopic behaviour of complex systems that present strong correlations among their elements, violating some of the essential properties of BG statistical mechanics. Such complex systems typically present power-law distributions, enhanced by (multi)fractal geometries, long-range interactions and/or large fluctuations between the various possible states, properties that correspond well to the collective behaviour of earthquakes and faults. Here, we provide an overview on the fundamental properties and applications of NESP. Initially, we provide an overview of the collective properties of earthquake populations and the main empirical statistical models that have been introduced to describe them. We provide an analytic description of the fundamental theory and the models that have been derived within the NESP framework to describe the collective properties of earthquakes. The fundamental laws of Statistical seismology as that of Gutenberg-Richter (GR) and Omori law a analysed using the ideas of Tsallis entropy and its dynamical superstatistical interpretation offered by Beck and Cohen.