Nonstandard mixing in the standard map

The standard map is a paradigmatic one-parameter (noted a) twodimensional conservative map which displays both chaotic and regular regions. This map becomes integrable for a = 0. For a 6= 0 it can be numerically shown that the usual, Boltzmann-Gibbs entropy S1(t) = − ∑ i pi(t) ln pi(t) exhibits a linear time evolution whose slope hopefully converges, for very fine graining, to the Kolmogorov-Sinai entropy. However, for increasingly small values of a, an increasingly large time interval emerges, before that stage, for which linearity with t is obtained only for the generalized nonextensive entropic form Sq(t) = 1− ∑ i[pi(t)] q q−1 with q = q ≃ 0.3. This anomalous regime corresponds in some sense to a power-law (instead of exponential) mixing. This scenario might explain why in isolated classical long-range N-body Hamiltonians, and depending on the initial conditions, a metastable state (whose duration diverges with 1/N → 0) is observed before it crosses over to the BG regime. PACS numbers: 05.20.-y, 05.45.-a, 05.70.Ce In his critical remarks about the domain of validity of Boltzmann principle, Einstein stressed [1] that the basis of statistical mechanics lies on dynamics. Intensive work has recently been done which is consistent with this standpoint, specifically in situations where anomalous effects may arise ([2, 3, 4, 5, 6, 7]). Also, a particularly interesting observation was made in [8], where it was found a simple connection between the Kolmogorov-Sinai (KS) entropy rate (the one that stems from the properties of mixing of the system) and the statistical entropy (the one that originates from a probability distribution). It was shown in fact that, partitionning the phase space in W cells and starting from many points within one cell, the time dependence of the usual, Boltzmann-Gibbs (BG), statistical entropy