Nonstandard mixing in the standard map

2001/08/29 by Fulvio Baldovin, Constantino Tsallis, Baldovin, Fulvio +3
Physics and Astronomy · #FOS: Physical sciences #Statistical Mechanics (cond-mat.stat-mech) #cond-mat.stat-mech

paper · pdf · doi:10.48550/arxiv.cond-mat/0108501

latex, 11 pages, 5 figures

arxiv created 2001/09/19 · arxiv updated 2009/11/30

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 ≠ 0 it can be numerically shown that the usual, Boltzmann-Gibbs entropy S₁(t)=-∑_i p_i(t)lnp_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)=\frac1-∑_i[p_i(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.

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