Next: 3.3.4 Markov Chains (Biased Up: 3.3 Random Sequences Previous: 3.3.2 Generating a stationary

3.3.3 Wiener-Lévy Process (Unbiased Random Walk)

With in the above differential equation, we find

where is Gaussian with

Since and are uncorrelated, we have

EXAMPLE: Let be one cartesian coordinate of a diffusing particle. Then is the mean squared displacement after time steps. In this case we may relate the coefficient to the diffusion constant according to .

EXERCISE: 500 random walkers set out from positions homogeneously distributed in the interval . 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 . Sketch the particle density after 100, 200, ... steps.

It is not really necessary to draw from a Gaussian distribution. If comes from an equidistribution in , the compound'' -increment after every steps will again be Gauss distributed (central limit theorem).

We may even discretize the -axis: with equal probability : after many steps, and on a scale which makes 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.

Next: 3.3.4 Markov Chains (Biased Up: 3.3 Random Sequences Previous: 3.3.2 Generating a stationary
Franz J. Vesely Oct 2005