Theory and phenomenology of growth oscillations in simple deterministic models

Abstract

We study growth oscillations in a class of deterministic models of growth which may be regarded as mean-field-like approximations to certain more realistic stochastic models. We demonstrate that the general morphology of the stochastic and deterministic models are similar. We then show that growth oscillations produce a rich multiperiodic structure in the deterministic models, as they do in the related stochastic systems. In both cases the multiperiodic oscillations are due to an induced incommensuration between a fundamental length scale (in this case the lattice spacing) and a dynamical length scale (in this case, the average distance grown per time step). We show how to estimate the effects of the growth oscillations and how to solve the deterministic equations analytically. This solution leads to a series of approximations, successive terms of which include the effects of longer-wavelength contributions to the multiperiodic growth oscillations. Finally, we present a sequence of approximations, the first one of which is the deterministic model studied here, which approach the related stochastic models.