Nonextensive Statistical Mechanics

Chapter 5Deterministic Dynamical Foundationsof Nonextensive Statistical MechanicsIl dépend de celui qui passe, Que je sois tombe ou trésor, Queje parle ou me taise, Ceci ne tient qu’à toi, Ami n’entre passans désir.Paul Valéry, Palais de ChaillotIn this chapter, we focus on microscopic-like nonlinear dynamical systems, inthe sense that the time evolution is expressed exclusively with deterministic ingredients.We will first discuss, analytically and numerically, low-dimensional dissipativemaps, and then low-dimensional conservative maps. We address next, numerically,many-body problems, first symplectic systems constituted by coupled simplelow-dimensional conservative maps, and finally classical Hamiltonian systems. Ourintention is to focus, in an unified manner, on those common aspects which relate tononextensive statistical mechanical concepts. We shall see that, every time we havenonlinear dynamics which is only weakly chaotic (typically at the frontier betweenregular motion and strong chaos), the need systematically emerges to q-generalizevarious concepts and functions, and very especially the entropy.5.1 Low-Dimensional Dissipative Maps5.1.1 One-Dimensional Dissipative MapsLet us start by defining the maps we are going to deal with. We focus on unimodalone-dimensional maps. Some of them are since long well known in the literature;others have been recently introduced with the purpose of illustrating specific featuresthat we are interested in.The z-logistic map is defined as follows (see, for instance, [128]:x t+1 = 1 − a|x t | z (z > 1; 0 ≤ a ≤ 2; |x t |≤1) . (5.1)The standard case is recovered for z = 2, and its primary edge of chaos occurs ata c (2) = 1.40115518909 ...For this simple z = 2 case, and with y ≡ x + 1/2, weobtain the traditional formC. Tsallis, Introduction to Nonextensive Statistical Mechanics,DOI 10.1007/978-0-387-85359-8 5, C○ Springer Science+Business Media, LLC 2009151