2016/12/12 by Guiomar Ruiz, Ugur Tirnakli, Ernesto P. Borges +1 · 1 citation
Physics and Astronomy · #cond-mat.stat-mech
paper · pdf · doi:10.1088/1742-5468/aa728b
20 pages, 21 figures
arxiv created 2016/12/12 · arxiv updated 2017/06/28
The standard map, paradigmatic conservative system in the (x,p) phase space, has been recently shown to exhibit interesting statistical behaviors directly related to the value of the standard map parameter K. A detailed numerical description is achieved in the present paper. More precisely, for large values of K, the Lyapunov exponents are neatly positive over virtually the entire phase space, and, consistently with Boltzmann-Gibbs (BG) statistics, we verify qent=qsen=qstat=qrel=1, where qent is the q-index for which the nonadditive entropy Sq ≡ k \frac1-∑i=1W piqq-1 (with S1=SBG ≡ -k∑i=1W pi ln pi) grows linearly with time before achieving its W-dependent saturation value; qsen characterizes the time increase of the sensitivity ξ to the initial conditions, i.e., ξ∼ e_qsen^λ_qsen t (λ_qsen>0), where eqz ≡[1+(1-q)z]1/(1-q); qstat is the index associated with the qstat-Gaussian distribution of the time average of successive iterations of the x-coordinate; finally, qrel characterizes the qrel-exponential relaxation with time of the entropy S_qent towards its saturation value. In remarkable contrast, for small values of K, the Lyapunov exponents are virtually zero over the entire phase space, and, consistently with q-statistics, we verify qent=qsen=0, qstat ≃ 1.935, and qrel ≃1.4. The situation corresponding to intermediate values of K, where both stable orbits and a chaotic sea are present, is discussed as well. The present results transparently illustrate when BG or q-statistical behavior are observed.