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| Main Author: | |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2506.02948 |
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Table of Contents:
- 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 $β$-FPUT system by removing the non-resonant terms. We work in the kinetic limit $N\to \infty$ and $β\to 0$ under the scaling laws $β=N^{-γ}$ with $0<γ<1$. The result holds up to the sub-kinetic time scale $T=N^{-ε}\min\bigl(N,N^{5γ/4}\bigr)=N^{-ε}T_{\mathrm{kin}}^{5/8}$ for $ε\ll 1$, 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.