Statistical Mechanics for Complex Systems

Boltzmann-Gibbs (BG) statistical mechanics constitutes, together with Newtonian mechanics, Einstein relativity (special and general), quantum mechanics, and Maxwell electromagnetism, a pillar of contemporary theoretical physics. It is grounded on the Boltzmann-Gibbs-von Neumann-Shannon entropic functional S_BG, which is additive. This magnificent theory provides, for 150 years, impressive successes (theoretical, experimental, observational, and computational) in a plethora of physical systems, more precisely those whose relevant space-time correlations are local (short-ranged). It fails, however, when they are nonlocal (long-ranged), which is the case of wide classes of natural, technological, and social complex systems. Nonadditive entropic functionals such as S_q (1988), S_delta (2009), S_{q,delta} (2013), Kaniadakis S_kappa (2001), Hanel-Thurner S_{c,d} (2011), and others, generalize S_BG and systematically overcome the BG difficulties. The basic idea is to violate the additivity of S_BG in such a way that the Legendre structure of classical thermodynamics remains preserved; indeed, for special values of the entropic indices (q, delta, kappa, c, d), the entropies corresponding to those classes of systems are thermodynamically extensive. Such generalizations consistently yield important theoretical advances such as Beck-Cohen superstatistics, properly scaled thermodynamics, nonlinear/inhomogeneous Fokker-Planck equations, generalized Central Limit Theorem and Large Deviations Theory, Einstein requirement for factorizability of the likelihood function, theory of critical phenomena and that of conservative and dissipative nonlinear dynamical systems at the edge of chaos, high-energy collisions (LHC, Brookhaven), asymptotically scale-free networks, quantum entanglement and quantum information, global optimization algorithms, learning machines, AI, signal and image processing, cosmological issues (black holes, dark energy), among others. The applications are falsifiable in the Popper sense and span from long-range-interactions in many-body Hamiltonian classical systems to those where dissipation (overdamping, type II superconductors) plays a crucial role, from cold atoms in optical lattices, spin glasses, and granular matter to quantum-tunneling chemical reactions and turbulence, from civil engineering and computational systems to astrophysical (IceCube Neutrino Observatory, Planck Observatory, Voyager 1/NASA,  AMS-02, James Webb Space Telescope) and geophysical (earthquakes, El Niño, ozone layer) ones, from economics, cancer, pandemics (COVID), and neurological diseases (Parkinson, ADHD, epilepsy) to human sciences such as linguistics. The aim of this master class is to operationally introduce this theory, generalized along the Boltzmann-Gibbs-von Neumann-Shannon legacy, through its foundations and most typical illustrations.

Textbook: C. Tsallis, Introduction to Nonextensive Statistical Mechanics-Approaching a Complex World – Second Edition (Springer-Nature, 2023)

General Bibliography: https://tsallis.cbpf.br/biblio.htm

DIS master class participation is by invitation only.