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 $d_{ij}^{-α}$ ($α\ge 0$), $\{d_{ij}\}$ 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 $0 \le α/d < 1$ (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 $0 \le α/d < 1$.

6 pages including 5 figures