Fractal Properties of “Disordered Surfaces” and The Termite Problem
In this second lecture, I shall turn to the topic of fractal surfaces, for which knowledge of only “the” fractal dimension df is not sufficient to describe the essential physics. In fact, we must distinguish between the entire external perimeter or “hull,” with its corresponding fractal dimension dh, and the unscreened portion of the external perimeter and its fractal dimension d. A very simple argument gives rise to an expression for du in terms of df and dp, the fractal dimension of the particles used to probe the surface. We shell see that the concept of the unscreened perimeter is at the heart of transport in a two-component random medium made up of superconductors and normal resistors. In particular, the exponent \(\tilde s = s/v\) for characterizing the divergence of the macroscopic conductivity at the percolation threshold pc is identically equal to du.
KeywordsFractal Dimension Percolation Threshold Random Walk Model Percolation Cluster Cayley Tree
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