Most studies on the problem of equilibration of the Fermi-Pasta-Ulam-Tsingou (FPUT) system have focused on equipartition of energy being attained amongst the normal modes of the corresponding harmonic system. In the present work, we instead discuss the equilibration problem in terms of local variables, and consider initial conditions corresponding to spatially localized energy. We estimate the time-scales for equipartition of space localized degrees of freedom and find significant differences with the times scales observed for normal modes. Measuring thermalization in classical systems necessarily requires some averaging, and this could involve one over initial conditions or over time or spatial averaging. Here we consider averaging over initial conditions chosen from a narrow distribution in phase space. We examine in detail the effect of the width of the initial phase space distribution, and of integrability and chaos, on the time scales for thermalization. We show how thermalization properties of the system, quantified by its equilibration time, defined in this work, can be related to chaos, given by the maximal Lyapunov exponent. Somewhat surprisingly we also find that the ensemble averaging can lead to thermalization of the integrable Toda chain, though on much longer time scales.

中文翻译：

---

$$\alpha $$-FPUT 链中局部观测值的热化

大多数关于 Fermi-Pasta-Ulam-Tsingou (FPUT) 系统平衡问题的研究都集中在相应谐波系统的正常模式之间获得的能量均分。在目前的工作中，我们改为根据局部变量讨论平衡问题，并考虑对应于空间局部能量的初始条件。我们估计了空间局部自由度均分的时间尺度，并发现与正常模式观察到的时间尺度存在显着差异。在经典系统中测量热化必然需要一些平均，这可能涉及初始条件或时间或空间平均。在这里，我们考虑对从相空间中的窄分布中选择的初始条件进行平均。我们详细研究了初始相空间分布的宽度以及可积性和混沌对热化时间尺度的影响。我们展示了系统的热化特性，由其在这项工作中定义的平衡时间量化，如何与最大李雅普诺夫指数给出的混沌相关。有点令人惊讶的是，我们还发现整体平均可以导致可积 Toda 链的热化，尽管时间尺度要长得多。