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

Abstract

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}^{-α} (α≥0), {d_{ij}} being the distances between N oscillator sites. Starting from random initial conditions we compute the maximal Lyapunov exponent λ_{max} as a function of N. Our N≫1 results strongly indicate that λ_{max} remains constant and positive for α/d>1 (implying strong chaos, mixing, and ergodicity), and that it vanishes like N^{-κ} for 0≤α/d<1 (thus approaching weak chaos and opening the possibility of breakdown of ergodicity). The suitably rescaled exponent κ exhibits universal scaling, namely that (d+2)κ depends only on α/d and, when α/d increases from zero to unity, it monotonically decreases from unity to zero, remaining so for all α/d>1. The value α/d=1 can therefore be seen as a critical point separating the ergodic regime from the anomalous one, κ playing a role analogous to that of an order parameter. This scaling law is consistent with Boltzmann-Gibbs statistics for α/d>1, and possibly with q statistics for 0≤α/d<1.