Ordering and finite-size effects in the dynamics of one-dimensional transient patterns.

Abstract

We introduce and analyze a general one-dimensional model for the description of transient patterns which occur in the evolution between two spatially homogeneous states. This phenomenon occurs, for example, during the Fr\'eedericksz transition in nematic liquid crystals. The dynamics leads to the emergence of finite domains that are locally periodic and independent of each other. This picture is substantiated by a finite-size scaling law for the structure factor. The mechanism of evolution towards the final homogeneous state is by local roll destruction and associated reduction of local wave number. The scaling law breaks down for systems of size comparable to the size of the locally periodic domains. For systems of this size or smaller, an apparent nonlinear selection of a global wavelength holds, giving rise to long-lived periodic configurations which do not occur for large systems. We also make explicit unsuitability of a description of transient pattern dynamics in terms of a few Fourier mode amplitudes, even for small systems with a few linearly unstable modes.