This course belongs to the series International School on Complexity, directed by A. Zichichi, and co-directed by G. Benedek, M. Gell-Mann, A. Rapisarda and C. Tsallis. It will be held at the Ettore Majorana Foundation and Centre for Scientific Culture in Erice, Italy during the period July 2-8, 2019.

Purpose of the Course

After more than 140 years of impressive success there is no reasonable doubt that the Boltzmann-Gibbs (BG) entropy is the correct one to be used for a wide and important class of physical systems, basically those whose (nonlinear) dynamics is strongly chaotic i.e., for classical systems with positive maximal Lyapunov exponent, which are mixing and ergodic. However, a plethora of physical complex systems exists for which such simplifying dynamical hypotheses are violated; typical examples are those for which the maximal Lyapunov exponent vanishes, leading to sub-exponential sensitivity to the initial conditions, which can of course occur in a variety of mathematical ways. Corresponding anomalies are found in a variety of quantum systems as well. In order to statistically describe the dynamics of such systems, various generalised forms of statistical mechanics have been proposed such as those using the nonadditive entropies S_q (where q is a real number which, for q=1, recovers the BG entropy), kappa distributions (also known as q-Gaussians, where kappa is simply related to q), superstatistical approaches, among various others. In the last decades, these new generalised statistical mechanical formalisms have found a large variety of very successful applications, even beyond the realm of physics. This course aims to cover the most recent analytical, experimental, observational and computational aspects and examples where these new extended formalisms have found fruitful applications.

Topics include, but are not limited to:

• Generalised Central Limit theorems;
• Generalised Large Deviation theory;
• Low-dimensional nonlinear conservative and dissipative dynamical systems near the edge of chaos;
• Long-range-interacting many-body classical Hamiltonian systems;
• Complex networks;
• Area-law-like quantum systems;
• Applications in astrophysics, space and other plasma physics, geophysics, high energy physics, cosmology, granular matter, cold atoms, econophysics, theoretical and structural chemistry, biophysics, social systems, power grids, image and time series processing, among others.