Sensitivity to initial conditions of a d-dimensional long-range-interacting quartic Fermi-Pasta-Ulam model: Universal scaling

We introduce a generalized d-dimensional Fermi-Pasta-Ulam model in the presence of long-range interactions, and perform a first-principle study of its chaos for d = 1,2,3 through large-scale numerical simulations. The nonlinear interaction is assumed to decay algebraically as d(ij)(-alpha) (alpha >= 0), {d(ij)} being the distances between N oscillator sites. Starting from random initial conditions we compute the maximal Lyapunov exponent lambda(max) as a function of N. Our N >> 1 results strongly indicate that lambda(max) remains constant and positive for alpha/d > 1 (implying strong chaos, mixing, and ergodicity), and that it vanishes like N (kappa) for 0 <= alpha/d < 1 (thus approaching weak chaos and opening the possibility of breakdown of ergodicity). The suitably rescaled exponent kappa exhibits universal scaling, namely that (d + 2)kappa depends only on alpha/d and, when alpha/d increases from zero to unity, it monotonically decreases from unity to zero, remaining so for all alpha/d > 1. The value alpha/d = 1 can therefore be seen as a critical point separating the ergodic regime from the anomalous one, kappa playing a role analogous to that of an order parameter. This scaling law is consistent with Boltzmann-Gibbs statistics for alpha/d > 1, and possibly with q statistics for 0 <= alpha/d < 1.