Diffusion as a random walk

Figure 2.17. The lattice of one dimensional sites for a random walker. The sites are separated by distance D and the walker is allowed to be only at a site on the lattice, but in general we may interpret position as the walker being within the distance D/2 of a site.

We close this chapter with a discussion of diffusion - first a model of how the process takes place (this section), then diffusion with linear population growth (next section), and finally diffusion with nonlinear population growth. This material is merely an introduction (although it does help cement many of the ideas we've discussed thus far) and Chapters 7 and 8 will be an extensive study of diffusion processes and their applications to stochastic population theory. We will also use occasional references to rules of probability (material from Chapter 3).

We envision a ''random walker'' whose position is denoted by X(t). Exactly what is intended by the word ''walking'' does not matter at this point - X(t) could equally be the position of an individual in physical space, the frequency of a genotype, the size of a population, or the price of a stock. We assume that the walk takes place on a one dimensional lattice of ''sites'' (Figure 2.17) that are spaced distance D apart. We will ultimately let D shrink to 0, but not just yet. The sites will be indexed by the letter i and we thus measure distance by x = i D; for concreteness, we refer to X(t) as the position of the walker at time t. The walker will make moves (''jumps'') at a fixed time interval r, which will also be allowed to shrink to 0 later in this section. These jumps are characterized by a transition function p(s), which is the probability that if the walker is currently at the site i, it will next be at the site j a distance s away. For example, if the walker were to move by flipping a fair coin (e.g. moving left by one step if heads comes up, right by one step if tails comes up), then p(— 1) = p(1) = 1/2 and p(s) = 0 for any other value of s. If the walker moved by reaching into a bag and pulling one of two dice, marked left or right, and then rolling the second die to determine how far to move, then it could move left 1,2, 3, 4, 5, or 6 sites each with probability 1/12 and right the same amounts with the same probability.

Given this framework, we define px, t) = Probability{X(t) = x}. This seems sensible enough, but there is actually a subtlety to it. When we talk about the walker ''being at a site,'' we actually mean within the vicinity of the site. That is, when we say the walker is at

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