Diffusion, Vibrations and Chemical Reactions

Our estimate shows us that diffusion along a random walk is abnormally slow, in the sense that the exponent 1/2 relating the mean squared displacement to the time (and the number of steps N, if the walker takes regular paces) is less than the classical value of 1. As the crow flies, the explorer covers a smaller distance for the same number of paces, and this distance deficit worsens as time goes by (the discrepancy between a law and a relation linear in t increases indefinitely). This can be understood from the structure of the fractal, which leads the walker into regions with fewer and fewer roads. The same phenomenon, although with a different exponent, would be observed for diffusion across a percolating cluster, or any other fractal structure. 

Formally speaking, we postulate the existence of a walk fractal exponent D, giving the time t required to cover a distance r by a relation of the type t(r), where (and it amounts to the same thing) 

For a random walk in a plane or in three dimensions, = 2 but for a random walk along the lines of another random walk, = 4. On a percolating cluster, at the threshold, lies between 2 and 4, which means that it will be explored a little more quickly than would the structure of a random walk, but a little more slowly than would a plane or a 3-dimensional space. The dead branches of the cluster are like traps the walker must escape from in order to make progress. However, if the walker is located on the backbone of the cluster, much better progress can be made than would be possible through the structure of a random walk because, at the threshold, the backbone of a spanning cluster is more string-like than the latter. 

Another way of observing the anomalous nature of diffusion on a fractal is to find out what becomes of the diffusion coefficient D t), defined from the classic Einstein relation between mean squared displacement and time  

In a normal space, such as a solution, (r (t)) increases linearly with time and D is indeed a constant. On a fractal, however, as the mean squared displacement increases more slowly, D is no longer a constant, but rather a decreasing function of time. On the structure of a random walk, since (r (t)) D decreases as 

---