Instabilities and relaxation to equilibrium in long-range oscillator chains

TL;DR

This paper investigates how long-range interactions in oscillator chains affect their stability and relaxation to equilibrium, revealing sporadic instabilities and differences from short-range models in energy transfer and relaxation times.

Contribution

It introduces a long-range extension of the FPU model and uncovers the sporadic nature of instabilities and their dependence on initial excitation amplitude.

Findings

Stronger localization in mode space for long-range FPU.

Instabilities occur in narrow amplitude intervals.

Relaxation to equilibrium is slower in short-range FPU.

Abstract

We study instabilities and relaxation to equilibrium in a long-range extension of the Fermi-Pasta-Ulam-Tsingou (FPU) oscillator chain by exciting initially the lowest Fourier mode. Localization in mode space is stronger for the long-range FPU model. This allows us to uncover the sporadic nature of instabilities, i.e., by varying initially the excitation amplitude of the lowest mode, which is the control parameter, instabilities occur in narrow amplitude intervals. Only for sufficiently large values of the amplitude, the system enters a permanently unstable regime. These findings also clarify the long-standing problem of the relaxation to equilibrium in the short-range FPU model. Because of the weaker localization in mode space of this latter model, the transfer of energy is retarded and relaxation occurs on a much longer time-scale.