7 - Diffusion in loopless structures

Summary

Nature abounds with types of structures for which loops may be neglected. The simplest example are perhaps linear polymers – modeled by self-avoiding walks – but also branched polymers (modeled by lattice animals), DLA aggregates, trees and tree-like structures, river systems, networks of blood vessels, and percolation clusters (in d ≥ 6) are common examples.

Diffusion in loopless structures is a lot simpler than that in other disordered substrates, for which loops cannot be neglected, and it therefore yields itself to a more rigorous analysis. Chiefly, anexact relation between dynamical exponents (the walk dimension and spectral dimension) and structural exponents (the fractal dimension and chemical length exponent) may be derived.

Diffusion in combs is a reasonable model for diffusion in some random substrates: the delay of a random walker caused by dangling ends and bottlenecks may be well mimicked by the time spent in the teeth of a comb. This case can be successfully analyzed with a CTRW and other techniques.

Loopless fractals

A large class of fractals are tree-like in structure. They are characterized by the absence of loops (or loops are so scarce that they may be neglected). In Figs. 7.1 and 7.2 we show examples of deterministic loopless fractals. For the study of transport properties it is useful to define their backbone, or skeleton. It consists of the union of all shortest (chemical) paths connecting the root of the tree with the peripheral sites.