Effective Theories of Diffusion

Ultimately, diffusion is an atomic-level process characterized by the jiggling motions of atoms within their local energy wells, now and then punctuated by 

The Random Walk. The most compelling discrete effective theory of diffusion is that provided by the random walk model. This picture of diffusion is built around nothing more than the idea that the diffusing entities of interest exercise a series of uncorrelated hops. The key analytic properties of this process can be exposed without too much difficulty and will serve as the basis of an interesting comparison with the Fourier methods we will undertake in the context of the diffusion equation. 

If we are to consider the accumulated excursion made by a random walker in a succession of N hops the F of which is characterized by the hop vector r , this excursion may be written as the vector sum 

We imagine that each hop is of a magnitude a (i.e. the lattice parameter) and for the purposes of simplicity, will further constrain our analysis to a simple cubic lattice in three dimensions. We are interested in the mean excursion made by the walker in N steps, namely, 

As was noted above, each hop is assumed to be independent of its predecessor and is just as likely to be in the forward direction as it is along the reverse direction and hence (r ) = 0. For the three-dimensional simple cubic lattice (spanned by the three basis vectors ei, e2, es ) of interest here, what we are claiming is that each of the six possible hops that may emanate from a given site has an equal probability of 1/6. Hence, the average (r ) is given by (ei — ei + C2 — e2 -b 63 — es) = 0. As a result, we must seek a measure of the mean excursion which is indifferent to the forward-backward symmetry. In particular, we note that the mean-square displacement 

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