Chaos suppression in the large size limit for long-range systems

We consider the class of long-range Hamiltonian systems first introduced by Anteneodo and Tsallis and called the alpha -XY model. This involves N classical rotators on a d-dimensional periodic lattice interacting all to all with an attractive coupling whose strength decays as r(-alpha), r being the distance between sites. Using a recent geometrical approach, we estimate for any d-dimensional lattice the scaling of the largest Lyapunov exponent (LLE) with N, as a function of a in the large energy regime where rotators behave almost freely. We find that the LLE vanishes as N-kappa, with kappa = 1/3 for 0 less than or equal to alpha /d less than or equal to 1/2 and kappa = 2/3(1 - alpha /d) for 1/2 less than or equal to alpha /d < 1. These analytical results present a nice agreement with numerical results obtained by Campa et al, including deviations at small N.