# Universal route to thermalization in weakly-nonlinear one-dimensional chains

## TL;DR

This paper uses Wave Turbulence theory to show that weakly nonlinear one-dimensional chains universally thermalize over time due to resonances, with the equipartition time scaling as a power-law of nonlinearity strength, supported by numerical simulations.

## Contribution

It demonstrates the universal role of resonances in thermalization of nonlinear chains and extends the analysis to the thermodynamic limit with numerical validation.

## Key findings

- Resonances cause irreversible energy transfer among modes.
- All considered systems thermalize at large times.
- Equipartition time scales as a power-law with nonlinearity strength.

## Abstract

We apply Wave Turbulence theory to describe the dynamics on nonlinear one-dimensional chains. We consider $\alpha$ and $\beta$ Fermi-Pasta-Ulam-Tsingou (FPUT) systems, and the discrete nonlinear Klein-Gordon chain. We demonstrate that resonances are responsible for the irreversible transfer of energy among the Fourier modes. We predict that all the systems thermalize for large times, and that the equipartition time scales as a power-law of the strength of the nonlinearity. Our methodology is not limited to only these systems and can be applied to the case of a finite number of modes, such as in the original FPUT experiment, or to the thermodynamic limit, i.e. when the number of modes approach infinity. In the latter limit, we perform state of the art numerical simulations and show that the results are consistent with theoretical predictions. We suggest that the route to thermalization, based only on the presence of exact resonance, has universal features. Moreover, a by-product of our analysis is the asymptotic integrability, up to four wave interactions, of the discrete nonlinear Klein-Gordon chain.