The Aubry-Mather theorem for driven generalized elastic chains
Abstract
We consider uniformly (DC) or periodically (AC) driven generalized infinite elastic chains (a generalized Frenkel-Kontorova model) with gradient dynamics. We first show that the union of supports of all the invariant measures, denoted by A, projects injectively to a dynamical system on a 2-dimensional cylinder. We also prove existence of ergodic invariant measures supported on a set of rotationaly ordered configurations with an arbitrary (rational or irrational) rotation number. This shows that the Aubry-Mather structure of ground states persists if an arbitrary AC or DC force is applied. The set A attracts almost surely (in probability) configurations with bounded spacing. In the DC case, the set A consists entirely of equilibria and uniformly sliding solutions. The key tool is a new weak Lyapunov function on the space of translationally invariant probability measures on the state space, which counts intersections.
Cite
@article{arxiv.1305.1109,
title = {The Aubry-Mather theorem for driven generalized elastic chains},
author = {Siniša Slijepčević},
journal= {arXiv preprint arXiv:1305.1109},
year = {2013}
}
Comments
27 pages, 1 figure