Random Walk

The so-called random walk is a stochastic process that governs, for example, the path traced by a molecule as it travels through a liquid or a gas, while constantly colliding with the other molecules in the medium. (See Section 5.3.9.)

Consider a random walk in one dimension. Suppose that a molecule takes steps of equal length $l$ along the $x$-axis. Suppose, further, that the steps are taken to the left (i.e., in the negative $x$-direction) or to the right, at random, with equal probabilities. Let $x_n$ be the molecule's $x$ coordinate after $n$ steps. It is assumed that $x_0=0$. In other words, the molecule is initially at the origin. We can write

$$x_n = x_{n-1}\pm l.$$ (5.43)

Hence,

$$x_n^{\,2} = x_{n-1}^{\,2} \pm 2\,x_n\,l+l^{\,2},$$ (5.44)

which implies that

$$\left\langle x_n^{\,2}\right\rangle = \left\langle x_{n-1}^{\,2}\right\rangle + l^{\,2}.$$ (5.45)

Thus, by induction, after $N$ steps, we obtain

$$\left\langle x^{2}\right\rangle = N\,l^{\,2}$$ (5.46)

Suppose that the steps are taken at a mean frequency $f$. It follows that $N=f\,t$, where $x=0$ at $t=0$. Hence,

$$\left\langle x^{2}\right\rangle = 2\,D\,t,$$ (5.47)

where

$$D =\frac{1}{2}\, f\,l^{\,2}$$ (5.48)

is known as the diffusivity. According to Equation (5.47), the molecule's mean square distance from its starting point grows linearly in time. This type of motion is known as diffusion. (See Section 5.3.9.)

Consider a random walk in three dimensions. Let ${\bf r}$ be the displacement of our molecule from the origin (which is its starting point). Suppose that the molecule takes steps of uniform length $l$, in a random direction, $f$ times a second. Let ${\bf l}$ be the displacement associated with a given step. Let ${\bf r}_n$ be the molecule's displacement after $n$ steps. We can write

$${\bf r}_n = {\bf r}_{n-1} + {\bf l}.$$ (5.49)

Thus,

$$r_n^{\,2} = ({\bf r}_{n-1}+{\bf l})\cdot ({\bf r}_{n-1}+{\bf l}) = r_{n-1}^{\,2} + 2\,{\bf r}_{n-1}\cdot{\bf l} + l^{\,2}.$$ (5.50)

However, if ${\bf l}$ is in a random direction then $\langle {\bf r}_{n-1}\cdot{\bf l} \rangle =0$, because the cosine of the angle subtended between ${\bf r}_{n-1}$ and ${\bf l}$ is just as likely to be positive as to be negative. Hence, the average of the previous equation yields

$$\left\langle r_n^{\,2}\right\rangle = \left\langle r_{n-1}^{\,2}\right\rangle +l^{\,2}.$$ (5.51)

By induction, after $N$ steps, we obtain

$$\left\langle r^2\right\rangle = N\,l^{\,2},$$ (5.52)

which implies that

$$\left\langle r^{2}\right\rangle = 2\,D\,t,$$ (5.53)

where $D=(1/2)\,f\,l^{\,2}$. Thus, the motion of the molecule is again diffusive in nature.