A measure of criticality for a complex system in the context of non-extensive statistical mechanics

Probability distributions with power-law behaviours are observed in many complex systems, reflecting hierarchical or fractal structures. Statistical mechanics provides descriptions of these systems, also through approaches which generalize the classical concept of entropy. In particular, we focused on the nonextensive Tsallis entropy which has been applied in many research fields and, in this work, is used for the study of earthquake sequences. The maximization of Tsallis entropy under suitable constraints yields to the Tsallis q-exponential distribution; the value of its parameter q characterizes the statistical properties of the system. We derive the expression of the q-exponential distribution which controls the behavior of seismic energy, which is related to earthquake magnitude; other geophysical variables can be considered, for instance length of faults and inter-event times. The q parameter is then estimated by following a Bayesian approach in order to identify different dynamical regimes in seismic sequences.