Diffusion on a DLA cluster in two and three dimensions

Synthesis of Lotus-Leaf-Shaped and Four-Footed ZnO Nanostructure on a Large Scale

“…d [16] e [17] f [2] g [3] for random walks or other statistical systems on fractals, the above discussion provides a basis to relate geometric properties to ν w (or D w ), with a reasonable precision, in structures with similar D F . It may guide investigations on models for real self-similar structures.…”

“…They are parts of the structures where the walk is forced to return over its own path. Among those fractals, dead ends occur in two-dimensional percolation clusters [14,15], DLA [16,17] and the T -fractal [4]. Each of these fractals may be compared to the other ones with similar D F (see table 2), which do not have this property and clearly have greater ν w .…”

Generalization of the Fractal Einstein Law Relating Conduction and Diffusion on Networks

“…Note that the spectral dimension [19], and hence ζ , are site independent, even though ρ * can vary from site to site ifd > 2 . Equation (2) is exact for the fractal trees depicted in Table I shows available simulated data [9] for the properties of Eden trees [20,21] and DLA clusters [10,13]. The resistance of loopless fractals is proportional to the length of the shortest path ℓ between two sites which scales [9] as ℓ ∼ R d min ( so ζ = d min ).…”

“…Although there is preponderance of evidence in their favour, the exactness of both formulae is controversial [8,9]: computations on two important fractals appear to violate the laws. Jacobs et al [13] have used simulation to show that three-dimensional DLA has d ≈ 1.35 , whereas 2d f /d w ≈ 1.55. Furthermore, their results predict ζ = d w −d f ≈ 0.71, which disagrees with the estimated value [9,14] ζ ≈ 1.…”

“…DLA 2D 1.20 ± 0.05 [13] 2.64 ± 0.05 [13] 1.70 ± 0.02 [13] 1.00 ± 0.02 [10] 0.94 ± 0.07 1.05 ± 0.09 DLA 3D 1.35 ± 0.05 [13] 3.19 ± 0.08 [13] 2.48 ± 0.02 [13] 1.02 ± 0.03 [10] 0.71 ± 0.10 1.04 ± 0.10…”