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##

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*

See also: "Computational Physics - An Introduction," Kluwer-Plenum 2001