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 (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 $0 \le \alpha/d < 1$ (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 $0 \le \alpha/d < 1$.