On the Hierarchy of Scales in Modeling of Weakly Interacting Chains of Atoms

Abstract

In the first part of this paper, we apply a well known discrete-to-continuum approach to a Frenkel-Kontorova-type model of an infinitely long one-dimensional chain of atoms weakly interacting with a line of fixed atoms. The rescaled model contains a small parameter $\delta$ that is the ratio of the strengths of the weak interaction and the elastic interaction. After replacing discrete displacements with piecewise affine functions to define continuum versions of the discrete energies, we prove that these energies $\Gamma$-converge to a continuum energy as $\delta\rightarrow 0$. This limiting process represents a transition from the microscale, at which individual atoms are resolved, to a mesoscale with a single diffuse domain wall. In the second part of this paper, we introduce an additional rescaling $\varepsilon$, and an associated limiting process that converts our problem to the macroscale. The $\varepsilon$-limiting energy is finite for piecewise constant functions of bounded variation. In the context of our problem, each point of discontinuity of a minimizer of the limiting energy corresponds to a sharp domain wall.