Biomedical Engineering Reference
In-Depth Information
statistical randomness to allow for longer linear displacement steps, with the condi-
tion that they remain small compared to the free space defined by the surrounding
geometry.
5.4.1.1 Two-Dimensional Case
In the two-dimensional case, a particle moves in a time step D
t
from the location (
x,
y
) to the location (
x
+ D
x
,
y
+ D
y
), where the space increments are defined by
D =
x
4
D t
D
cos( )
α
(5.48)
D =
y
4
D t
D
sin( )
α
α
=
random
(0,2 )
π
In (5.48) the function “random” is a choice of uniformly distributed random
numbers in the interval [0, 2
p
]. The validity of the method depends on the quality
of randomness of the angle
a
. In the following we have used the Matlab command
“rand” [11]. The length of the displacement has been scaled by the real diffusion
length scale.
Example of Random Walk from a Source Point
We show here two examples of random walk calculations. The first case is the 2D
diffusion from a source point (Figure 5.24).
Using the algorithm defined by (5.48), we find the images in Figure 5.25. Diffu-
sion is isotropic around the initial spot and has a Gaussian shape along a radius.
In Figure 5.26, it can be verified that the average square distance is related to
the time by the relation
2
<
d
> =
4
Dt
Example of Random Walk in a Microchannel
The same algorithm can be applied to the case of a microchannel. The results are
shown in Figure 5.27. In this case also, the particle distribution follows a Gaussian
profile.
Figure 5.24
2D diffusion of tracers originated at a source point.
