The effect of long-range interactions on the dynamics and statistics of 1D Hamiltonian lattices with on-site potential

TL;DR

This study investigates how long-range interactions influence the dynamics and statistical properties of 1D Hamiltonian lattices with on-site potentials, revealing unique low-dimensional solutions and distinct chaos indicators.

Contribution

It demonstrates the impact of long-range interactions on discrete breathers, Lyapunov exponents, and momentum distributions in two specific 1D lattice models, highlighting differences from short-range systems.

Findings

Single-site excitations form low-dimensional solutions described by Duffing oscillators.

Lyapunov exponent behavior varies with energy density and system size, indicating different chaos regimes.

Momentum distributions under LRI differ significantly from FPU model, showing non-Gaussian features.

Abstract

We examine the role of long--range interactions on the dynamical and statistical properties of two 1D lattices with on--site potentials that are known to support discrete breathers: the Klein--Gordon (KG) lattice which includes linear dispersion and the Gorbach--Flach (GF) lattice, which shares the same on--site potential but its dispersion is purely nonlinear. In both models under the implementation of long--range interactions (LRI) we find that single--site excitations lead to special low--dimensional solutions, which are well described by the undamped Duffing oscillator. For random initial conditions we observe that the maximal Lyapunov exponent $\lambda $ %: (a) tends to a positive value for KG and grows like $\varepsilon^(.25)$ for GF as the energy density $\varepsilon=E/N$ increases; (b) saturates to a positive value as the number of particles $N$ increase, scales as $N^{-0.12}$ in the KG model and as $N^{-0.27}$ in the GF with LRI, suggesting in that case an approach to integrable behavior towards the thermodynamic limit. Furthermore, under LRI, their non-Gaussian momentum distributions are distinctly different from those of the FPU model.