强非线性晶格中的遍历与非遍历多体动力学
Ergodic and non-ergodic many-body dynamics in strongly nonlinear lattices
摘要：
经典多体动力学中非线性振子链的研究有着悠久的历史，可以追溯到Fermi，Pasta，Ulam和Tsingou（FPUT）的开创性工作。我们介绍了一个新的家庭这样的系统，其中包括链的$N$谐波耦合粒子的非线性引入限制每个单独的粒子的运动到一个盒子/体育场与硬墙。视距被布置在一维晶格上，但是它们单独地不必是一维的，从而允许已经在晶格尺度上引入混沌。在大多数情况下，我们研究的情况下，运动是完全一维的。我们发现，该系统表现出一个混合相空间的任何有限值的$N$。在随机挑选的相空间位置和哈密顿演化和相空间平均值之间的直接比较的计算的Lyapunov谱表明，相空间的规则区域是不显着的大系统尺寸。虽然我们模型的连续极限本身就是可积sinh-Gordon理论的一个奇异极限，但我们没有看到FPUT工作中著名的非遍历性的任何证据。最后，我们研究了链与粒子局限于二维体育场的个人体育场已经混乱，并发现一个更混乱的相空间在小系统的大小。
英文摘要：
The study of non-linear oscillator chains in classical many-body dynamics has a storied history going back to the seminal work of Fermi, Pasta, Ulam and Tsingou (FPUT). We introduce a new family of such systems which consist of chains of $N$ harmonically coupled particles with the non-linearity introduced by confining the motion of each individual particle to a box/stadium with hard walls. The stadia are arranged on a one dimensional lattice but they individually do not have to be one dimensional thus permitting the introduction of chaos already at the lattice scale. For the most part we study the case where the motion is entirely one dimensional. We find that the system exhibits a mixed phase space for any finite value of $N$. Computations of Lyapunov spectra at randomly picked phase space locations and a direct comparison between Hamiltonian evolution and phase space averages indicate that the regular regions of phase space are not significant at large system sizes. While the continuum limit of our model is itself a singular limit of the integrable sinh-Gordon theory, we do not see any evidence for the kind of non-ergodicity famously seen in the FPUT work. Finally, we examine the chain with particles confined to two dimensional stadia where the individual stadium is already chaotic, and find a much more chaotic phase space at small system sizes.
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