Statistical mechanics for complex systems: Foundations and applications

Boltzmann-Gibbs entropy and statistical mechanics is one of the pillars of contemporary physics. It applies extremely successfully to the so called simple systems, essentially ergodic. When we wish to study complex systems, particularly nonergodic ones, a more powerful theory is needed. For a wide class of such complex systems, nonadditive entropies and the associated statistical mechanics are being currently used and studied. Recent aspects related to its foundations and applications are now available. A brief overview will be presented. Foundations concern nonlinear dynamics, central limit theorems, large deviation theory, probabilistic correlations, calculation of the index q from first principles, among others. Applications concern long-range-interacting many-body classical Hamiltonian systems (e.g., XY rotator and Fermi-Pasta-Ulam models), over-damped motion of repulsively interacting vortices in type-II superconductors, high energy physics (e.g., distributions of momenta in high energy collisions at CERN/LHC, Brookhaven/RHIC), granular matter (position fluctuations in two-dimensional shear motion), plasma physics (e.g., distributions of velocities), financial laws (e.g., distributions of price returns and of interoccurrence times)geophysics (seismic analyses of geophysical areas in Greece and elsewhere), biology (chemical distances between classes of nucleotides in DNA sequences of modern and archaic bacteria and Homo Sapiens), cold atoms, image and signal processing, among others. A bibliography is available at http://tsallis.cat.cbpf.br/biblio.htm