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The BCF theory of crystal growth has been successful in describing a widerange of phenomena in surface physics. Typical crystal surfaces are slightlymisoriented with respect to a facet plane; thus, the BCF theory views suchsystems as composed of staircase-like structures of steps separating terraces.Adsorbed atoms (adatoms), which are represented by a continuous density,diffuse on terraces, and steps move by absorbing or emitting these adatoms.Here we shed light on the microscopic origins of the BCF theory by deriving asimple, one-dimensional (1D) version of the theory from an atomistic, kineticrestricted solid-on- solid (KRSOS) model without external material deposition.We define the time-dependent adatom density and step position as appropriateensemble averages in the KRSOS model, thereby exposing the non-equilibriumstatistical mechanics origins of the BCF theory. Our analysis reveals that theBCF theory is valid in a low adatom-density regime, much in the same way thatan ideal gas approximation applies to dilute gasses. We find conditions underwhich the surface remains in a low-density regime and discuss the microscopicorigin of corrections to the BCF model.

From atoms to steps: The microscopic origins of crystal evolution