Biharmonic pattern selection.

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 ∇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 or 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 rl ≈ 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.