STATISTICAL MECHANICS OF SYSTEMS LYING OUTSIDE THE DOMAIN OF VALIDITY OF THE BOLTZMANN-GIBBS THEORY

Description
The celebrated statistical mechanics introduced by Boltzmann and Gibbs more than one century ago lie (for classical systems, for instance) on hypotheses such as ergodicity and mixing. Strongly chaotic systems, with positive maximum Lyapunov exponent, satisfy requirements of this sort. Within this realm, relevant random variables are probabilistically independent or nearly so. It is for such situations, and related quantum ones, that the central limit theorem and the standard entropy (Boltzmann, Gibbs, von Neumann, Shannon) exhibit their well known utility and connections with classical thermodynamics. What can be done outside this world? Can we approach such anomalous, and nevertheless ubiquitous, cases on thermostatistical grounds similar to the usual ones? For wide classes of such systems the answer appears to be positive, by appropriately generalizing the entropy and, consistently, the central limit theorem. Some central concepts as well as typical predictions, verifications and applications for natural, artificial and social systems will be briefly presented. BIBLIOGRAPHY: (i) C. Tsallis, Entropy, in Encyclopedia of Complexity and Systems Science (Springer, Berlin, 2009); (ii) C. Tsallis, Introduction to Nonextensive Statistical Mechanics - Approaching a Complex World (Springer, New York, 2009); (iii) S. Umarov, C. Tsallis, M. Gell-Mann and S. Steinberg, J. Math. Phys. 51, 033502 (2010); (iv) CMS Collaboration, J. High Energy Phys. 02, 041 (2010); (v) http://tsallis.cat.cbpf.br/biblio.htm
Organised by Andrea Rapisarda