Nonextensive Statistical Mechanics

180 5 Deterministic Dynamical Foundations of Nonextensive Statistical Mechanicstorus (mod 1). Additionally, the maps are placed in a one dimensional (d = 1)regular lattice with periodic boundary conditions. The distance r ij is the minimumdistance between maps i and j, hence it can take values from unity to N 2( N−12 )for even (odd) number N of maps. Note that r ij is a fixed quantity that, modulatedwith the power α, enters Eq. (5.35) as an effective time-independent couplingconstant. As a consequence, α regulates the range of the interaction betweenmaps. The sum is global (i.e., it includes every pair of maps), so the limiting casesα = 0 and α =∞correspond, respectively, to infinitely long range and nearestneighbors. In our case d=1, thus 0 ≤ α ≤ 1(α > 1) means long-range (shortrange)coupling. Moreover, the coupling term is normalized by the sum [177, 360]Ñ ≡ d ∫ N 1/d1dr r d−1 r −α = N 1−α/d −α/d, to yield a non-diverging quantity as the systemsize grows (for simplicity, we have replaced here the exact discrete sum over1−α/dinteger r by its continuous approximation).If G(¯x) denotes a map system, then G is symplectic when its Jacobian G/¯xsatisfies the relation [83]:( ) G T ( ) GJ = J , (5.36)¯x ¯xwhere the superindex T indicates the transposed matrix, and J is the Poisson matrix,defined by( ) 0 IJ ≡ , (5.37)−I 0I being the N × N identity matrix. A consequence of Eq. (5.36) is that the Jacobiandeterminant |G/¯x| =1, indicating that the map G is (hyper)volume-preserving.In particular, for our model( )G I I¯x = , (5.38)B (I + B)where ¯x is the 2N-dimensional vector ¯x ≡ ( ¯p, ¯θ), and⎛⎞K θ1 c 21 ... c N1c 12 K θ2 ... c N2B = ⎜⎝...⎟. ⎠ , (5.39)c 1N c 2N ... K θNwithK θi≡ a cos[2πθ i (t)] + b Ñ∑j≠icos[2π(θ i (t) − θ j (t))]rijα , (5.40)andc ij = c ji ≡− b Ñcos[2π(θ i (t) − θ j (t))]rijα , (5.41)