Statistical characterization of the standard map - CSH

Statistical characterization of the standard map


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The standard map, paradigmatic conservative system in the (xp) phase space, has been recently shown (Tirnakli and Borges (2016 Sci. Rep6 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 $q_{\rm{ent}}$ , the sensitivity to initial conditions $q_{\rm{sen}}$ , the distribution of a time-averaged (over successive iterations) phase-space coordinate $q_{\rm{stat}}$ , and the relaxation to the equilibrium final state $q_{\rm{rel}}$ , collapse onto a fixed point, i.e. $q_{\rm{ent}}=q_{\rm{sen}}=q_{\rm{stat}}=q_{\rm{rel}}=1$ .

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 $q_{\rm{ent}}=q_{\rm{sen}}=0$ $q_{\rm{stat}} \simeq 1.935$ , and $q_{\rm{rel}} \simeq1.4$ . 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.

 

G. Ruiz, U. Tirnakli, E.P. Borges, C. TsallisStatistical characterization of the standard map, J. Stat. Mech. (2017) 063403