Nonextensive Statistical Mechanics

5.3 High-Dimensional Conservative Maps 179Fig. 5.30 Time dependence of S q averaged over the 10% quick-best spreading cells, on the farfrom-equilibriumregime, for typical values of q.In order to have a finite entropy production for S q we need a value of q which isdefinitively smaller than unity, i.e., the Boltzmann–Gibbs entropy does not appearas the most adequate tool. Deeper studies are needed in order to establish whetheranother value of q can solve this problem.5.3 High-Dimensional Conservative MapsThe model we focus on here [359] is a set of N symplectically coupled (henceconservative) standard maps, where the coupling is made through the coordinatesas follows:θ i (t + 1) = θ i (t) + p i (t + 1) (mod 1),p i (t + 1) = p i (t) + a2π sin[2πθ i(t)]+b2π ÑN∑j=1j≠isin[2π(θ i (t)−θ j (t))]r α ij(mod 1),(5.35)where t is the discrete time t = 1, 2,..., and α ≥ 0. The a parameter is the usualnonlinear constant of the individual standard map, whereas the b parameter modulatesthe overall strength of the long-range coupling. Both parameters contributeto the nonlinearity of the system; it becomes integrable when a = b = 0. Forsimplicity, we have studied only the cases a > 0, b > 0, but we expect similarresults when one or both of these parameters are negative. The systematic study ofthe whole parameter space is certainly welcome. Notice that, in order to describea system whose phase-space is bounded, we are considering, as usual, only the