Frenkel-Kontorova模型非公度结构的动力学

Dynamics of incommensurate structures of the Frenkel-Kontorova model

本文研究Frenkel-Kontorova(FK)模型非公度结构的动力学.所谓非公度结构是指粒子的平均间距是无理数的情形,此时系统的相空间是无穷维的.本文证明在过阻尼条件下,系统在无穷维相空间中有一个一维流形,关于Poincar′e映射和空间变换群是不变的,并且此二者在此流形上均诱导出圆周上的保向同胚.二者的旋转数分别关联于系统的平均速度和粒子的平均间距.

In this paper, we study the dynamics of incommensurate structures of the Frenkel-Kontorova model.When the mean spacing of the particles is irrational, we say that the system has incommensurate structures, for which the phase space of the system is infinite-dimensional. We show that under the overdamped condition,there exists a one-dimensional manifold which is invariant both for the Poincar′e map and space shifts. Moreover,the Poincar′e map and space shifts induce orientation-preserving circle homeomorphisms on this manifold with rotation numbers connecting respectively with the average velocity and the mean spacing.