A diffusion model for transport on loopless aggregates

We propose a diffusion equation for transport on a loopless aggregate. The relevant distance variable is the chemical distance and the spatially dependent coefficients of the diffusion equation are found in terms of B(I), the number of bonds at chemical distance I from the vertex of the tree, and Bs(I), the number of bonds on the skeleton of the tree. It is shown that the resulting motion can be modelled in terms of a biased one-dimensional random walk with a probability of remaining in place that approaches 1 as I → ∞. This is interpreted as being due to motion along dead ends. An important consequence of the model is a simple derivation of the expressions relating the diffusion exponent and fracton dimension to the fractal dimension and the intrinsic dimensions of the tree and its skeleton. The mean first-passage time to go from the vertex to an arbitrary shell is also found.