Rigorous Derivation of the Wave Kinetic Equation for $β$-FPUT System
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866913889970552832 |
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| author | Wu, Boyang |
| author_facet | Wu, Boyang |
| 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_02948 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Rigorous Derivation of the Wave Kinetic Equation for $β$-FPUT System Wu, Boyang Analysis of PDEs Mathematical Physics 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. |
| title | Rigorous Derivation of the Wave Kinetic Equation for $β$-FPUT System |
| topic | Analysis of PDEs Mathematical Physics |
| url | https://arxiv.org/abs/2506.02948 |