Harmonic Measure and Passivation of 2D and 3D Fractals

Denis Grebenkov
École Polytechnique
Laboratoire de Physique de la Matiere Condensee

Many transport processes in nature and industry are governed by diffusion and thus subject to Laplacian screening. In the first part of the talk, we introduce a fast random walk algorithm adapted to self-similar geometrical structure of deterministic Von Koch boundaries in 2D and 3D. Modeling Brownian motion by this technique enables us to calculate the distribution of arrival probabilities (harmonic measure) for high order generations of these curves and surfaces. In the second part of the talk, we discuss the multifractal properties of the harmonic measure on the Von Koch boundaries in 2D and 3D. In particular, the information dimension for the cubic Von Koch surface is computed and shown to be slightly greater than 2. The third part of the talk concerns a numerical study of the passivation of irregular interfaces. At each passivation step, the most accessible region of the interface is passivated that is, the Dirichlet boundary condition is turned into the Neumann boundary condition. The evolution of the successive accessible regions is investigated numerically for quadratic and cubic Von Koch boundaries. In sharp contrast with 2D case (for which the length of these regions remains approximately constant), their surface area progressively decreases in 3D.

Presentation (PDF File)

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