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

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 [Formula: see text] 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 [Formula: see text]-converge to a continuum energy as [Formula: see text]. 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 [Formula: see text], and an associated limiting process that converts our problem to the macroscale. The [Formula: see text]-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.