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

6 pages including 5 figures

References in corpus (8)

• Superdiffusion and non-Gaussian statistics in a driven-dissipative 2D dusty plasma
• Power-Law Distributions for a Trapped Ion Interacting with a Classical Buffer Gas
• Thermostatistics of overdamped motion of interacting particles
• The standard map: From Boltzmann-Gibbs statistics to Tsallis statistics
• Experimental validation of nonextensive scaling law in confined granular media
• Fermi-Pasta-Ulam model with long-range interactions: Dynamics and thermostatistics
• Universal threshold for the dynamical behavior of lattice systems with long-range interactions
• Controlling the Range of Interactions in the Classical Inertial Ferromagnetic Heisenberg Model: Analysis of Metastable States

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• The effect of long-range interactions on the dynamics and statistics of 1D Hamiltonian lattices with on-site potential
• Heat transport in long-ranged anharmonic oscillator models
• Scaling properties of d-dimensional complex networks
• Statistical Entropy of Open Quantum Systems
• Approaching a large deviation theory for complex systems
• Non-Gaussian Saha's ionization in Rindler spacetime and the equivalence principle
• Effective chemical potential and its phenomenological implications for the Hubble parameter