Laplacian growth of parallel needles: A Fokker-Planck equation approach

Using a conformal transformation to set up the iterative nonlinear equations, We Study analytically the kinetics of growth of parallel needles. We establish a discrete Fokker-Planck equation for the probability of finding at time t a given distribution of needle lengths. In the linear regime, it shows a short-wavelength Laplacian instability which we investigate in detail. From the crossover of the solutions to the nonlinear regime, we deduce analytically the general scale invariance of the two-dimensional models.