3.3.3 Wiener-Lévy Process (Unbiased Random Walk)

With $\beta=0$ in the above differential equation, we find

$$x(n+1) = x(n)+z(n)$$

where $z(n)$ is Gaussian with

$$z(n) \equiv \int_{0}^{\Delta t} s(t_{n}+t')\,dt' \;,\;\;\;\; \langle z \rangle =0 \; , \;\;\;\; \langle z^{2} \rangle =A\Delta t$$

Since $z$ and $x$ are uncorrelated, we have

$$\langle [x(n)]^{2} \rangle = n\,A\,\Delta t$$

EXAMPLE: Let $x$ be one cartesian coordinate of a diffusing particle. Then $\langle [x(n)]^{2} \rangle$ is the mean squared displacement after $n$ time steps. In this case we may relate the coefficient $A$ to the diffusion constant according to $A=2D$.






EXERCISE: 500 random walkers set out from positions $x(0)$ homogeneously distributed in the interval $[-1,1]$. The initial particle density is thus rectangular. Each of the random walkers is now set on its course to perform its own one-dimensional trajectory, with $A\,\Delta t = 0.01$. Sketch the particle density after 100, 200, ... steps.


It is not really necessary to draw $z(n)$ from a Gaussian distribution. If $z(n)$ comes from an equidistribution in $[-\Delta x/2,\; \Delta x/2]$, the ``compound'' $x$-increment after every $10-15$ steps will again be Gauss distributed (central limit theorem).

We may even discretize the $x$-axis: $z=0, \; \pm \Delta x$ with equal probability $1/3$: after many steps, and on a scale which makes $\Delta x$ appear small, the results will again be the same.

To simulate 2- or 3-dimensional diffusion, apply the above procedure independently to 2 or 3 coordinates.