A model to describe fractal growth is introduced that includes effects due to long-range coupling. The model is based on the biharmonic equation ${\mathrm{\nabla }}^4$u=0 in two-dimensional isotropic defect-free media as follows from the Kuramoto-Sivashinsky equation for patter formation–or, alternatively, from the theory of elasticity. As a difference with alternative Laplacian and Poisson growth models, in this model the Laplacian of u is neither zero nor proportional to u. Its discretization allows one 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 ${\mathit{r}}_{\mathit{l}}$≊L/${\mathit{e}}^{1/2}$ 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 analyzed.

• Received 16 September 1992

DOI:https://doi.org/10.1103/PhysRevE.47.1243