Nonstandard mixing in the standard map
Abstract
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 \ne 0 it can be numerically shown that the usual, Boltzmann-Gibbs entropy S_1(t)=-\sum_{i} p_i(t)\ln{p_i(t)} exhibits a {\it 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, {\it before} that stage, for which {\it linearity} with t is obtained only for the generalized nonextensive entropic form S_q(t)=\frac{1-\sum_{i}[p_i(t)]^{q}}{q-1} with q = q^*\simeq 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\to 0) is observed before it crosses over to the BG regime.