Nonextensive statistical mechanics, anomalous diffusion and central limit theorems (Q862032)

This paper is in part a review of the work by the author on nonextensive statistical mechanics which he initiated in 1988 with subsequent refinements in 1991 and 1998. The paper also contains some new results regarding the anomalous diffusion and various (thus far unsuccessful) attempts at proving the central limit theorem within the context of his nonextensive statistical mechanics. Although not stated explicitly in the text, the rationale for nonextensivity in some models of statistical mechanics is discussed in Ref. [4]. In this paper the author provides all definitions from scratch thus making it accessible to nonexperts. In addition to extending the well established concept of the entropy within the scope of equilibrium-type problems, some illustrative nonequilibrium applications to problems such as computations of anomalous diffusion, Lyapunov exponents, etc. extending works by \textit{Ya. B. Pesin} [Dimension theory in dynamical systems: contemporary views and applications. Chicago Lectures in Mathematics. Chicago: The University of Chicago Press (1998; Zbl 0895.58033)] is also presented.