Statistical characterization of the standard map

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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 $q_{\rm{ent}}$ , 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. Tsallis, Statistical characterization of the standard map, J. Stat. Mech. (2017) 063403

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Constantino Tsallis

CSH External Faculty