Sensitivity to initial conditions of \(d\)-dimensional long-range-interacting Fermi-Pasta-Ulam model: Universal scaling

Abstract

We introduce a generalized \(d\)-dimensional \(\beta\)-Fermi-Pasta-Ulam (FPU) model in 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 \(d_{ij}^{-\alpha}\) (\(\alpha \ge 0\)), \(\{d_{ij}\}\) 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 \(0 \le \alpha/d < 1\) (hence approaching weak chaos, thus 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, so remaining 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 \(q\)-statistics for \(0 \le \alpha/d < 1\).