The smoothed Laplacian model: A mean field approximation to Laplacian growth

Abstract

We introduce a new model for irreversible growth inspired by Laplacian growth processes. The growth probability is defined using an approximation for the Laplacian field which takes into account the screening effect only at the largest scale. For this reason the resulting cluster is not a fractal as those generated with diffusion limited aggregation but this makes the model fully soluble in a mean field approximation. The results are non-trivial and give a clear description of the stationary regime of growth which corresponds to the dynamical equilibrium between the growth probability and the evolving cluster. This model then provides a new perspective in the study of Laplacian growth phenomena, from which it is possible to make predictions on the behaviour of more complex systems. For example it is possible to recover in a simple way the correct value of the finger width in the Saffman-Taylor problem as the only one consistent with this equilibrium situation. It is also possible to understand the physical origin of the statistical properties which give rise to the breakdown of the multifractal spectrum for negative moments applied to the probability distribution of the dielectric breakdown model.