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 $q=q^{\ast }\simeq 0.3$. 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.

latex, 11 pages, 5 figures