Dimensionalities for the harmonic and ballistic measures of fractal aggregates.

Abstract

The fractal dimensionalities have been determined for a variety of random fractals in which the masses of occupied lattice sites or particles are weighted by their probabilities of being contacted by random walkers (harmonic measure) or by particles following linear trajectories (ballistic measure) which are absorbed after their first contact. Using the harmonic measure, fractal dimensionalities (${D}_{h}$) of 0.925\ifmmode\pm\else\textpm\fi{}0.05 are found for a variety of two-dimensional random structures with different fractal dimensionalities [percolation clusters (D\ensuremath{\simeq}1.89), Witten-Sander clusters (D\ensuremath{\simeq}1.70), cluster-cluster aggregates (D\ensuremath{\simeq}1.43), and screened-growth clusters (D=1.25, 1.50, and 1.75)]. Using the ballistic measure, fractal dimensionalities of 1.26\ifmmode\pm\else\textpm\fi{}0.02 were obtained for Witten-Sander aggregates and 1.17\ifmmode\pm\else\textpm\fi{}0.02 for percolation clusters. For screened-growth clusters with fractal dimensionalities of 1.25, 1.5, and 1.75, the fractal dimensionality with ballistic measure was found to be 1.25\ifmmode\pm\else\textpm\fi{}0.05, 1.31\ifmmode\pm\else\textpm\fi{}0.03, and 1.28\ifmmode\pm\else\textpm\fi{}0.03, respectively. Consequently, it seems that the fractal dimensionality associated with the harmonic measures is the same for all two-dimensional objects and has a value consistent with that predicted by the theoretical work of Grassberger.