Nonstandard mixing in the standard map

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The standard map is a paradigmatic one-parameter (noted a) two-dimensional conservative map which displays both chaotic and regular regions. This map becomes integrable for a=0. For a ≠ 0 it can be numerically shown that the usual, Boltzmann-Gibbs entropy S₁(t)=-∑ᵢ pᵢ(t)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)]qq-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.