Kinetic Hierarchies and Macroscopic Limits for Crystalline Steps in $1+1$ Dimensions

Abstract

We apply methods of kinetic theory to study the passage from particle systems to nonlinear partial differential equations (PDEs) in the context of deterministic crystal surface relaxation. Starting with the near-equilibrium motion of N line defects (“steps”) with atomic size a, we derive coupled evolution equations (“kinetic hierarchies”) for correlation functions $F_n^a$, which express correlations of n consecutive steps. We investigate separately the evaporation-condensation and the surface diffusion dynamics in $1+1$ dimensions when each step interacts repulsively with its nearest neighbors. In the limit $a\to0$ with $Na=\mathcal{O}(1)$, where a is appropriately nondimensional, the first equations of the hierarchies reduce to known evolution laws for the surface slope profile. The remaining PDEs take the form of simple continuity equations, which we solve exactly and, thereby, connect continuous limits of $F_n^a$ with the slope profile. In addition, we construct a particular example of $F_n^a$ asymptotically for small but finite a by regularization of measures. In the limit $a\to0$, this construction yields singular correlation functions.