  A theory based on the thermodynamic Gibbs-Thomson relation is presented which
provides the framework for understanding the time evolution of isolated
nanoscale features (i.e., islands and pits) on surfaces. Two limiting cases are
predicted, in which either diffusion or interface transfer is the limiting
process. These cases correspond to similar regimes considered in previous works
addressing the Ostwald ripening of ensembles of features. A third possible
limiting case is noted for the special geometry of "stacked" islands. In these
limiting cases, isolated features are predicted to decay in size with a power
law scaling in time: A is proportional to (t0-t)^n, where A is the area of the
feature, t0 is the time at which the feature disappears, and n=2/3 or 1. The
constant of proportionality is related to parameters describing both the
kinetic and equilibrium properties of the surface. A continuous time Monte
Carlo simulation is used to test the application of this theory to generic
surfaces with atomic scale features. A new method is described to obtain
macroscopic kinetic parameters describing interfaces in such simulations.
Simulation and analytic theory are compared directly, using measurements of the
simulation to determine the constants of the analytic theory. Agreement between
the two is very good over a range of surface parameters, suggesting that the
analytic theory properly captures the necessary physics. It is anticipated that
the simulation will be useful in modeling complex surface geometries often seen
in experiments on physical surfaces, for which application of the analytic
model is not straightforward.
