Deterministic Dynamical Foundations of Nonextensive Statistical Mechanics

In this chapter, we focus on microscopic-like nonlinear dynamical systems, in the sense that the time evolution is expressed exclusively with deterministic ingredients. We will first discuss, analytically and numerically, low-dimensional dissipative maps, and then low-dimensional conservative maps. We address next, numerically, many-body problems, first symplectic systems constituted by coupled simple low-dimensional conservative maps, and finally classical Hamiltonian systems. Our intention is to focus, in an unified manner, on those common aspects which relate to nonextensive statistical mechanical concepts. We shall see that, every time we have nonlinear dynamics which is only weakly chaotic (typically at the frontier between regular motion and strong chaos), the need systematically emerges to q-generalize various concepts and functions, and very especially the entropy.