The Aubry-Mather theorem for driven generalized elastic chains

TL;DR

This paper extends the Aubry-Mather theory to driven generalized elastic chains, demonstrating the persistence of ordered states and invariant measures under external forces, with implications for understanding their long-term dynamics.

Contribution

It proves the existence of invariant measures and the Aubry-Mather structure for driven elastic chains, including under arbitrary AC or DC forces, using a novel weak Lyapunov function.

Findings

Supports of invariant measures project injectively to a 2D cylinder

Existence of ergodic invariant measures with arbitrary rotation numbers

The set of supports attracts configurations with bounded spacing

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.