Biharmonic pattern selection

TL;DR

This paper introduces a biharmonic growth model based on the $ abla^4 u=0$ equation, capturing long-range interactions in fractal pattern formation, and analyzes the transition from dense to multibranched growth.

Contribution

It presents a novel biharmonic model for pattern growth, extending previous Laplacian-based models, and provides analytical estimates for transition points in circular geometries.

Findings

Transition from dense to multibranched growth occurs at a system size-dependent point.

In circular geometry, the transition point is approximately at 60% of the system radius.

Biharmonic patterns influence growth probabilities at lattice sites.

Abstract

A new model to describe fractal growth is discussed which includes effects due to long-range coupling between displacements $u$. The model is based on the biharmonic equation $\nabla^{4}u =0$ in two-dimensional isotropic defect-free media as follows from the Kuramoto-Sivashinsky equation for pattern formation -or, alternatively, from the theory of elasticity. As a difference with Laplacian and Poisson growth models, in the new model the Laplacian of $u$ is neither zero nor proportional to $u$. Its discretization allows to reproduce a transition from dense to multibranched growth at a point in which the growth velocity exhibits a minimum similarly to what occurs within Poisson growth in planar geometry. Furthermore, in circular geometry the transition point is estimated for the simplest case from the relation $r_{\ell}\approx L/e^{1/2}$ such that the trajectories become stable at the growing surfaces in a continuous limit. Hence, within the biharmonic growth model, this transition depends only on the system size $L$ and occurs approximately at a distance $60 \%$ far from a central seed particle. The influence of biharmonic patterns on the growth probability for each lattice site is also analysed.