Non-Extensive Statistical Mechanics: Overview of Theory and Applications in Seismogenesis, Climate, and Space Plasma

Abstract

In this small review, the theoretical framework of non-extensive statistical theory, introduced by Constantino Tsallis in 1988, is presented in relation with the q-triplet estimation concerning experimental time series from climate, seismogenesis, and space plasmas systems. These physical systems reveal common dynamical, geometrical, or statistical characteristics. Such characteristics are low dimensionality, typical intermittent turbulence multifractality, temporal or spatial multiscale correlations, power law scale invariance, non-Gaussian statistics, and others. The aforementioned phenomenology has been attributed in the past to chaotic or self-organized critical (SOC) universal dynamics. However, after two or three decades of theoretical development of the complexity theory, a more compact theoretical description can be given for the underlying universal physical processes which produce the experimental time series complexity. In this picture, the old reductionist view of universality of particles and forces is extended to the modern universality of multiscale complex processes from the microscopic to the macroscopic level of different physical systems. In addition, it can be stated that a basic and universal organizing principle exists creating complex spatio-temporal and multiscale different physical structures or different dynamical scenarios at every physical scale level. The best physical representation of the underline universal organizing principle is the well-known entropy principle. Tsallis introduced a q-entropy (S q ) as a non-extensive (q-extension) of the Boltzmann–Gibbs (BG) entropy (for q = 1, the BG entropy is restored) and statistics in order to describe efficiently the rich phenomenology that complex systems exhibit. Tsallis q-entropy could be a strong candidate for entropy principle according to which nature creates complex structures everywhere, from the microscopic to the macroscopic level, trying to succeed the extremization of the Tsallis entropy. In addition, this Sq entropy principle is harmonized with the q extension of the classic and Gaussian central limit theorem (q-CLT). The q-extension of CLT corresponds to the Levy a-stable extension of the Gaussian attractor of the classic statistical theory. The q-CLT is related to the Tsallis q-triplet theory of random time series with non-Gaussian statistical profile. Moreover Tsallis q-extended entropy principle can be used as the theoretical framework for the unification of some new dynamical characteristics of complex systems such as the spatio-temporal fractional dynamics, the anomalous diffusion processes and the strange dynamics of Hamiltonian and dissipative dynamical systems, the intermittent turbulence theory, the fractional topological and percolation phase transition processes according to Zelenyi and Milovanov non-equilibrium and non-stationary states (NESS) theory, as well as the non-equilibrium renormalization group theory(RNGT) of distributed dynamics and the reduction of dynamical degrees of freedom.