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

Assuming that resonances play a major role in the transfer of energy among the Fourier modes, we apply the 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 consider both the thermodynamic limit and the discrete regime and we conjecture that all the systems thermalize for large times, and that the equipartition time scales as a power-law of the strength of the nonlinearity, at least for a range of values of the nonlinear parameter. We perform state of the art numerical simulations and show that the results are mostly consistent with theoretical predictions. Some observed discrepancies are discussed. We suggest that the route to thermalization, based 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.