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Nanoscale materials hold the promise of leading to breakthroughs in the development of electronics. These materials are of great interest especially at low temperatures due to their thermal stability. In order to predict the evolution of crystal surfaces at such precision, physical effects across a wide range of scales, from atomistic processes to large-scale thermodynamics, must be consolidated.

This thesis aims to incorporate the microscale information carried by the atomic dynamics to the evolution of an apparently smooth surface at macroscopic scale. At the nanometer scale, the motion of atomic defects in the surface is described by ordinary differential equations (ODEs). At larger scale, the atomic roughness is no longer detectable and the surface evolution can be described by a smooth function for the surface height on some reference plane. This height function satisfies certain partial differential equations (PDEs) on the basis of the thermodynamic principles. These ODEs and PDEs separately yield predictions of distinct characteristics for the morphological evolution of a surface. While modeling at small scale has the advantage of simple physical principles, observation at the larger scale offers more tangible intuition for the topographic evolution and it is often more suitable for relating to experiments.

A principal theme of this thesis is to understand the difference or error between these two predictions. The error can be conveniently assessed numerically but this is not sufficient to achieve a deeper understanding of the problem. To this end, this thesis addresses both quantitative notion of the error through numerics and systematic and conceptual notion of the error. In order to give a concrete notion to this difference, it is crucial to carefully interpret what is meant by a solution of the evolusion PDEs; the subtlety pertains to the choice of method used to solve the PDE. Recently, it has been shown that the solutions of PDEs obtained solely from the thermodynamic principles are prone to deviate from the underlining microscopic dynamics. This thesis investigates the cause of this discrepancy and propose a reconciliation by exploring a new continuum model that may plausibly incorporate microscopic influences.