In the past few years the Nonlinear and Statistical Physics Research Group at Emory has made considerable advances in understanding nonequilibrium surface and interface growth phenomena through the development of new concepts and techniques in theoretical, simulational and experimental studies. In particular, the importance and the role of nonlinear dynamical processes, scaling and fractals in growth phenomena has been recognized.
Recently, however, it was shown that DLA in two dimensions can be grown by iterating stochastic conformal maps. The basic idea is to follow the evolution of the conformal mapping which maps the exterior of the unit circle in the mathematical--plane onto the complement of the cluster of $n$ particles in the physical --plane.
The equation of motion for this mapping is determined recursively: the cluster of n ``particles'' is created by adding a new ``particle'' of constant shape and linear scale to the cluster of (n-1) ``particles'' at a position which is chosen randomly according to the harmonic measure. Using such iterated stochastic conformal maps DLA can be generated (see Figure). Thus another important approach to understanding DLA will involve generalizations of this basic strategy to provide a new method for investigating of the spatiotemporal development of Laplacian flows in wedge shaped geometries. In addition, analysis of such iterated stochastic conformal maps can be used to investigate the evolution of the multifractal distribution of pressure gradients at the interface which generate the complex branched structures observed in DLA.
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( submitted Physical Review E, 2000).