Double scaling in the relaxation time in the $\beta$-FPUT model

TL;DR

This paper investigates the relaxation time in the $eta$-FPUT model, revealing a transition in scaling behavior linked to frequency overlap and resonant interactions, supported by numerical simulations and theoretical analysis.

Contribution

It introduces a simple formula for nonlinear frequency broadening and demonstrates a transition in thermalization time scaling related to Chirikov overlap criteria.

Findings

For very small nonlinearity, the relaxation time matches wave resonance theory.

A different scaling law emerges when frequency overlap occurs.

The Chirikov overlap criterion marks the transition between two relaxation regimes.

Abstract

We consider the original $\beta$-Fermi-Pasta-Ulam-Tsingou ($\beta$-FPUT) system; numerical simulations and theoretical arguments suggest that, for a finite number of masses, a statistical equilibrium state is reached independently of the initial energy of the system. Using ensemble averages over initial conditions characterized by different Fourier random phases, we numerically estimate the time scale of equipartition and we find that for very small nonlinearity it matches the prediction based on exact wave-wave resonant interactions theory. We derive a simple formula for the nonlinear frequency broadening and show that when the phenomenon of overlap of frequencies takes place, a different scaling for the thermalization time scale is observed. Our result supports the idea that Chirikov overlap criterium { identifies} a transition region between two different relaxation time scaling.