Nonextensive Statistical Mechanics

26 2 Learning with Boltzmann–Gibbs Statistical MechanicsS(A + B) = F(S(A), S(B); {η}) , (2.28)where F(x, y; {η}) is a function which, besides depending symmetrically on (x, y),depends on a (typically small) set of universal indices {η}. In other words, it doesnot depend on the microscopic configurations of A and B. Equivalently, we are ableto macroscopically calculate the entropy of the composed system without any needof entering into the knowledge of the microscopic states of the subsystems. Thisproperty appears to be a natural one for an entropic form if we desire to use it as abasis for a statistical mechanics which would naturally connect to thermodynamics.The BG entropy is composable since it satisfies Eq. (2.10). In other words, wehave F(x, y) = x + y. Since S BG is nonparametric, no index exists.2.1.2.10 Sensitivity to the Initial Conditions, Entropy Production per UnitTime, and a Pesin-Like IdentityFor a one-dimensional dynamical system (characterized by the variable x) the sensitivityto the initial conditions ξ is defined as follows:ξ ≡x(t)limx(0)→0 x(0) . (2.29)It can be shown [82, 83] that ξ paradigmatically satisfies the equationwhose solution is given bydξdt= λ 1 ξ, (2.30)ξ = e λ 1 t . (2.31)(The meaning of the subscript 1 will become transparent later on). If the Lyapunovexponent λ 1 > 0(λ 1 < 0), the system will be said to be strongly chaotic (regular).The case where λ 1 = 0 is sometimes called marginal and will be extensivelyaddressed later on.At this point let us briefly review, without proof, some basic notions of nonlineardynamical systems. If the system is d-dimensional (i.e., it evolves in a phase-spacewhose d-dimensional Lebesgue measure is finite), it has d Lyapunov exponents: d +of them are positive, d − are negative, and d 0 vanish, hence d + + d − + d 0 = d. Letus order them all from the largest to the smallest: λ (1)1≥ λ (2)1≥ ... ≥ λ (d +)1>λ (d ++1)1= λ (d ++2)1= ... = λ (d ++d 0 )1= 0 >λ (d ++d 0 +1)1≥ λ (d ++d 0 +2)1≥ ... ≥ λ (d)1 .An infinitely small segment (having then a well defined one-dimensional Lebesguemeasure) diverges like e λ(1) 1 t ; this precisely is the case focused in Eq. (2.31). An infinitelysmall area (having then a well defined two-dimensional Lebesgue measure)diverges like e (λ(1) 1 +λ(2) 1 ) t . An infinitely small volume diverges like e (λ(1) 1 +λ(2) 1 +λ(3) 1 ) t , andso on. An infinitely small d-dimensional hypervolume evolves like e [∑ dr=1 λ(r) 1 ] t .If