Evolution of crystal surfaces: From discrete schemes to continuum laws

Dionisios Margetis

In traditional settings such as fluids and classical elasticity the starting point (truth'') is often identified with continuum equations for macroscopic variables of interest (densities etc). But in many cases of mathematical modeling this perspective is changed: The truth is atomistic, or takes the form of discrete schemes, by which evolution laws must be determined and analyzed at the macroscale. In this talk I focus on the evolution of crystal surfaces as a prototypical case of two-scale modeling and analysis, with implications in the design of novel devices. The governing, discrete equations represent the motion of interacting line defects, steps'' with atomic height. In the appropriate limit a nonlinear PDE is derived for the surface height, and free-boundary problems are formulated for the surface motion. I show analytically how microscopic details of the crystal enter the requisite boundary conditions, and thus affect evolution at the macroscale.