Nonextensive Statistical Mechanics

3.2 Nonadditive Entropy S q 53whose solution is given bydξdt= λ q ξ q , (3.55)ξ = e λ q tq . (3.56)The paradigmatic case corresponds to λ q > 0 and q < 1. In this case we haveξ ∝ t 1/(1−q) (t →∞) , (3.57)(see also [122–126]) and we refer to it as weak chaos, in contrast to strong chaos,associated with positive λ 1 . To be more precise, Eq. (3.56) has been proved to be theupper bound of an entire family of such relations at the edge of chaos of unimodalmaps. For each specific (strongly or weakly) chaotic one-dimensional dynamicalsystem, we generically expect to have a couple (q sen ,λ qsen ) (where sen stands forsensitivity) such that we haveξ = e λ qsen tq sen. (3.58)Clearly, strong chaos is recovered here as the particular instance q sen = 1.Let us now address the interesting question of the S q entropy production astime t increases. By using S q instead of S BG , we could follow the same stepsalready indicated in Section 2.1.2, and attempt the definition of a q-generalizedKolmogorov–Sinai entropy rate. We will not follow along this line, but we shallrather q-generalize the entropy production K 1 introduced in Section 2.1.2. We definenowK q ≡ limlimlimt→∞ W →∞ M→∞S q (t)t. (3.59)We conjecture that generically an unique value of q exists, noted q ent (whereent stands for entropy) such that (the upper bound of) K qent is finite (i.e., positive),whereas K q vanishes (diverges) for q > q ent (q < q ent ).We further conjecture for one-dimensional systems thatand thatq ent = q sen , (3.60)K qent = K qsen = λ qsen . (3.61)As already mentioned, strong chaos is recovered as a particular case, and we obtainthe Pesin-like identity K 1 = λ 1 . Conjectures (3.58), (3.60), and (3.61) were firstintroduced in [127], and have been analytically proved and/or numerically verified