Propagation failure, depinning forces, and coupling parameters for particle chains

Abstract:

We utilize the Aubry-Mather theory to investigate the relation between coupling parameters and propagation failure intervals for particle chains. Assume $A<B$ are critical values such that the particle chain has three homogeneous equilibria provided the driving force $F\in (A,B)$. We obtain the following asymptotic relations: The propagation failure interval approaches to $[A,B]$ as the coupling parameter tends to $0$, and the propagation failure interval approaches to $\{0\}$ as the coupling parameter tends to $+\infty$.

The main tool of this paper is a physical quantity called the depinning force, which determines whether the state of the particle chain is pinning or sliding. Recently, it has been proved that the transition threshold $F_c^+$($F_c^-$) for the particle chain coincides with the upper(lower) limit of the upper(lower) depinning force as the rotation number approaches to zero from the right. Our investigations are based on this conclusion.