Statistical characterization of the standard map

The standard map, paradigmatic conservative system in the (x, p) phase space, has been recently shown (Tirnakli and Borges (2016 Sci. Rep. 6 23644)) to exhibit interesting statistical behaviors directly related to the value of the standard map external parameter K. A comprehensive statistical numerical description is achieved in the present paper. More precisely, for large values of K (e.g. K  =  10) where the Lyapunov exponents are neatly positive over virtually the entire phase space consistently with Boltzmann–Gibbs (BG) statistics, we verify that the q-generalized indices related to the entropy production $q_{ent}$, the sensitivity to initial conditions $q_{sen}$, the distribution of a time-averaged (over successive iterations) phase-space coordinate $q_{stat}$, and the relaxation to the equilibrium final state $q_{rel}$, collapse onto a fixed point, i.e. $q_{ent}=q_{sen}=q_{stat}=q_{rel}=1$. In remarkable contrast, for small values of K (e.g. K  =  0.2) where the Lyapunov exponents are virtually zero over the entire phase space, we verify $q_{ent}=q_{sen}=0$, $q_{stat}\simeq 1.935$, and $q_{rel}\simeq 1.4$. 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 behavior and/or q-statistical behavior are observed. (paper: interdisciplinary statistical mechanics)