Route to thermalization in the <i>α</i> -Fermi–Pasta–Ulam system

TLDR

The thermodynamic limit is briefly discussed. We study the α‑FPU system with 16–64 masses connected by nonlinear quadratic springs and analyze its long‑time dynamics using resonant wave‑wave interaction theory, assuming that exact resonances govern the weakly nonlinear regime. We find that six‑wave resonances drive energy thermalization in the α‑FPU system, with a predicted time scale of order ε⁻⁸, and that equipartition occurs for arbitrarily small nonlinearity, a result confirmed by extensive simulations and driven by Umklapp resonances.

Abstract

We study the original $\alpha$-Fermi-Pasta-Ulam (FPU) system with $N=16,32$ and $64$ masses connected by a nonlinear quadratic spring. Our approach is based on resonant wave-wave interaction theory, i.e. we assume that, in the weakly nonlinear regime (the one in which Fermi was originally interested), the large time dynamics is ruled by exact resonances. After a detailed analysis of the $\alpha$-FPU equation of motion, we find that the first non trivial resonances correspond to six-wave interactions. Those are precisely the interactions responsible for the thermalization of the energy in the spectrum. We predict that for small amplitude random waves the time scale of such interactions is extremely large and it is of the order of $1/\epsilon^8$, where $\epsilon$ is the small parameter in the system. The wave-wave interaction theory is not based on any threshold: equipartition is predicted for arbitrary small nonlinearity. Our results are supported by extensive numerical simulations. A key role in our finding is played by the {\it Umklapp} (flip over) resonant interactions, typical of discrete systems. The thermodynamic limit is also briefly discussed.