Rigorous Derivation of the Wave Kinetic Equation for $\beta$-FPUT System

TL;DR

This paper rigorously derives the wave kinetic equation for the $eta$-FPUT system in the kinetic limit, addressing long-standing questions about thermalization time scales and handling complex dispersion relations.

Contribution

It provides the first rigorous derivation of the wave kinetic equation for the $eta$-FPUT system, including non-polynomial dispersion relations and a phase renormalization technique.

Findings

Derivation valid up to sub-kinetic time scale $T=N^{-rac{ ext{small}}{1}}T_{ ext{kin}}^{5/8}$

Addresses non-polynomial dispersion relations in wave kinetic theory

Introduces phase renormalization to cancel divergent interactions

Abstract

Wave kinetic theory has been suggested as a way to understand the longtime statistical behavior of the Fermi-Pasta-Ulam-Tsingou (FPUT) system, with the aim of determining the thermalization time scale. The latter has been a major problem since the model was introduced in the 1950s. In this thesis we establish the wave kinetic equation for a reduced evolution equation obtained from the $\beta$-FPUT system by removing the non-resonant terms. We work in the kinetic limit $N\to \infty$ and $\beta\to 0$ under the scaling laws $\beta=N^{-\gamma}$ with $0<\gamma<1$. The result holds up to the sub-kinetic time scale $T=N^{-\epsilon}\min\bigl(N,N^{5\gamma/4}\bigr)=N^{-\epsilon}T_{\mathrm{kin}}^{5/8}$ for $\epsilon\ll 1$, where $T_{\mathrm{kin}}$ represents the kinetic (thermalization) timescale. The novelties of this work include the treatment of non-polynomial dispersion relations, and the introduction of a robust phase renormalization argument to cancel dangerous divergent interactions.