标题： 关于$β$-FPUT 系统的波动能方程的严格推导 显示英文标题

标题： Rigorous Derivation of the Wave Kinetic Equation for $β$-FPUT System

摘要： 波动能理论被提出作为一种理解费米-帕斯塔-乌拉姆-青柳（FPUT）系统长时间统计行为的方法，旨在确定热化的时间尺度。自20世纪50年代该模型被引入以来，后者一直是一个主要问题。在本论文中，我们为从$\beta$-FPUT系统通过去除非共振项得到的简化演化方程建立了波动能方程。我们在动力学极限$N\to \infty$和$\beta\to 0$下研究了遵循标度律$\beta=N^{-\gamma}$的情况，其中$0<\gamma<1$。 该结果在亚动力学时间尺度$T=N^{-\epsilon}\min\bigl(N,N^{5\gamma/4}\bigr)=N^{-\epsilon}T_{\mathrm{kin}}^{5/8}$内成立，对于$\epsilon\ll 1$，其中$T_{\mathrm{kin}}$表示动力学（热化）时间尺度。 本文的新颖之处包括处理非多项式色散关系，以及引入一种稳健的相位重新归一化论证来消除危险的发散相互作用。 显示英文摘要

摘要： 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.