Random Processes, Variance, and Diffusion

The degree of zone spreading for random processes depends on the length and frequency of the random steps and on the length of time over which they occur. A simplified model which describes the nature of this dependence is the random walk model [4,7,8]. 

Inasmuch as we have seen that the basic diffusion equations lead to a Gaussian profile, we may in most cases describe Gaussian spreading as an apparent diffusion process, with an effective diffusion coefficient related to r through Eq. 5.21 

If a random walk process underlies the apparent diffusion, D can be obtained in a precise form by substituting Eq. 5.24 into Eq. 5.25 

It is a fundamental statistical law that when several independent processes contribute to zone spreading, the variances are additive (4). Thus the total variance r2T is of the form 

It follows that the apparent diffusion coefficients, which Eq. 5.25 shows to be proportional to the respective variances, are additive in the same manner 

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RANDOM PROCESSES, VARIANCE, AND DIFFUSION 93 A physical rationale for this particular dependence [8] is summarized below. 

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