23.4 Continuous random walks (diffusion)

One final thought can be made here. We are taking discrete steps but we can transform our results to a continuous time analog. Let \(t= n \Delta t\) be the approximation from discrete time to continuous time. Equivalently \(\displaystyle n = \frac{t}{\Delta t}\). With this information we can rearrange the square displacement equation to the following:

\[\begin{equation} \big \langle (x^{n})^{2} \big \rangle = \frac{t}{\Delta t} ( \Delta x)^{2}. \end{equation}\]

The quantity \(\displaystyle D = \frac{( \Delta x)^{2}}{2 \Delta t}\) is known as the diffusion coefficent. So then the mean square displacement can be arranged as \(\langle (x^{n})^{2} \rangle = 2Dt\), confirming again that the variance grows proportional to \(t\).

To connect this back to our discussion of stochastics, understanding a random walk helps us to understand how demographic and environmental stochasticity affect a dynamical system and the types of behaviors in the solution this random walk introduces to the system. In the following sections we will investigate the ways in which the random walk connects to stochastic differential equations.