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

TL;DR

This paper presents a one-dimensional model for transient pattern dynamics in systems transitioning between homogeneous states, highlighting finite-size effects, local domain formation, and limitations of Fourier mode descriptions.

Contribution

It introduces a general model capturing finite-size effects and local domain formation during transient pattern evolution, revealing breakdown of scaling laws in small systems.

Findings

Finite-size scaling law for structure factor

Emergence of long-lived periodic configurations in small systems

Inadequacy of Fourier mode descriptions for transient dynamics

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 Freedericksz transition in nematic liquid crystals.The dynamics leads to the emergence of finite domains which 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 wavenumber. 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 the 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.