Nonextensive Statistical Mechanics

5.1 Low-Dimensional Dissipative Maps 155Broadhurst calculated the z = 2 Feigenbaum constant α F with 1018 digits [352].Through Eq. (5.12), it straightforwardly follows thatq sen (2) = 0.244487701341282066198 .... (5.13)See [128] for q sen (z).The same type of information is available for the edge of chaos of other unimodalmaps. For example, for the universality class of the z-circular map, we mustuse [132] b = ( √ 5 + 1)/2 = 1.6180 ... into Eq. (5.11). We then obtain [132]q sen (3) = 0.05 ± 0.01. Similar results are available for the universality class of thez-exponential map [146].5.1.1.3 Entropy Production and the Pesin TheoremThere are quite generic circumstances under which the entropy increases with time,typically while dynamically exploring the phase-space of the system. If this increaseis (asymptotically) linear with time we may define an entropy production per unittime, which is the rate of increase of the entropy. One such concept, based on singletrajectories as already mentioned, is the so-called Kolmogorov–Sinai entropy rate orjust KS entropy [84]. It satisfies, under quite general conditions, an identity, namelythat it is equal to the sum of all positive Lyapunov exponents (which reduces tothe single Lyapunov exponent if the system is one-dimensional). This equality isfrequently referred in the literature as the Pesin identity, orthePesin theorem [86].Here, instead of the KS entropy (computationally very inconvenient), we shall useK q , the ensemble-based entropy production rate that we defined in Section 3.2. Werefer to Eq. (3.59). A special value of q, noted q ent , generically exists such thatK qent is finite, whereas K q vanishes (diverges) for any q > q ent (q < q ent ). Forsystems strongly chaotic (i.e., whose single Lyapunov exponent is positive), we haveq ent = 1, thus recovering the usual case of ergodic systems and others. For systemsweakly chaotic (i.e., whose single Lyapunov exponent vanishes, such as in the caseof an edge of chaos), we have q ent < 1. Many nonergodic (but certainly not all)systems belong to this class.For quite generic systems we expect [127] (see Section 5.2)andq ent = q sen , (5.14)K qent = λ qsen . (5.15)For q ent = 1, this entropy production is expected to coincide, quite generically,with the KS entropy rate. Although a rigorous proof is, to the best of our knowledge,still lacking, examples can be seen in [133, 139, 147]. For many K 1 = λ 1 = 0