Random Walk Simulation
Introduction of Random Walk Simulation
Gaussian Random Walk in one dimension space
The process simulates the random walk of particle in the original point (0,0) along y
axis. The step of every motion (∆y) is produced using the function of random (‘Normal’, 0,
3) in matlab. Here, ‘Normal’ represents Gaussian (Normal) distribution, 0 is the mean
value, and 3 is the standard deviation. Here are some examples of random data (red lines
represent the fit curves).
NOTE: Here, we decide the mean value as 0 because we want to make probabilities of
particle’s motion along forward and backward direction to be equal.
Figure 1. Histograms of random data with Gaussian distribution and fit curves.
Figure 2 is the sketch diagram. The particle moves along the y axis and every step
length (∆y) is random data with regards of Gaussian distribution.
y
∆y
x
Figure 2. Random walk in one dimension space. ∆y data are relative to the Gaussian
distribution.
Gaussian Random Walk in two dimension space
1
In the two dimension space, we should consider both ∆x and ∆y. These data are also
random and relative to Gaussian distribution like above, respectively. The detailed process
is as following. There is a particle in the original point. Now, let it to walk in the two
dimension space. Every time, the particle firstly walks a random length (∆y) along y axis,
and then walks another random length (∆x) along x axis. In other words, the random walk
in the two dimension space is the combination of two separate random walks in one
dimension space. The sketch of the random walk in two dimension space is shown in the
figure 3.
y
∆x
∆y
x
Figure 3. Random walk in two dimension space. Both ∆x and ∆y data are relative to
the Gaussian distribution. The Random walk in two dimension space is combination
of two random walks along x and y axis, respectively.
Gaussian Random Walk in a circular orbital
Here, we let the particle to walk randomly along a circular orbital with radius of 3. We
only produce one group of random data (∆l) using function of random (‘Normal’, 0, 3).
Relatively, we define, when random data are negative, the particle will anticlockwise walk
a length along the orbital; in contrast, when random data are plus, the particle will
clockwise walk along the orbital. Every time, the length of the arc is equal to the value of
random datum. Figure 4 shows the sketch of the random walk in a circular orbital with
radius of 3.
y
3
∆l1
∆l2
-3
3 x
-3
Figure 4. Random walk in a circular orbital with radius of 3. The length of arc (∆l) is
relative to the Gaussian distribution. When the particle moves anticlockwise, ∆l is
negative; when the particle moves clockwise, ∆l is plus.
2
NOTE: For every different space, we choose different steps to simulate the random
walk, and these steps are 10, 50, 200, 1000, 5000, and 10000, respectively. For every
chosen number of total steps, we simulate it for four times. For examples, in the one
dimension space, we simulate the random walk for four times with total steps of 10000.
Besides, we assume 30 step frames per second. In other words, every step takes 0.033
seconds. This is convenient to draw the autocorrelation curves.
Trajectory Pictures of Random Walk Simulation
In one dimension space
Step=10
Step=50
Step=200
Step=1000
Step=5000
Step=10000
Figure 5. Trajectories of Random walk in one dimension space with different steps.
3
In two dimension space
Step=10
Step=50
Step=200
Step=1000
Step=5000
Step=10000
Figure 6. Trajectories of Random walk in two dimension space with different steps.
In a circular orbital with radius of 3
Step=10
Step=50
Step=200
Step=1000
Step=5000
Step=10000
Figure 7. Trajectories of Random walk in a circular orbital (radius = 3) with
different steps.
4
Autocorrelation Analysis
In one dimension space
Step=10
Step=50
Step=200
Step=1000
Step=5000
Step=10000
Figure 8. Autocorrelation curves of Random walk in one dimension space with
different steps
In two dimension space
Step=10
Step=50
Step=200
Step=1000
Step=5000
Step=10000
Figure 9. Autocorrelation curves of Random walk in two dimension space with
different steps
5
In a circular orbital with radius of 3
Step=10
Step=50
Step=200
Step=1000
Step=5000
Step=10000
Figure 10. Autocorrelation curves of Random walk in a circular orbital (radius = 3) with
different steps.
Conclusion:
From the autocorrelation curves, we can see, when the steps of simulation are few, the
autocorrelation curves have large noises. However, the curves looks like smooth lines with
time increasing in the figures with larger number of simulation steps for random walk in all
three different spaces. Besides, the curves decay down to zero quickly when times increase
from zero. According to the definition, autocorrelation function can show the relationship
of the velocities, indicating the property of motion. Thus, the curves above show there is no
relationship for random walk in three different spaces between the velocities in different
time. This is in agreement with the process of production of random data, because there is
no relationship between two random numbers.
6
Degree statistical Analysis
Rotation angle:
Position 2
 (1)
 (1)
 (2)
Position 4
Position 1
Position 3
 (2)
_
+




NOTE: As shown in the picture above, we define the sign of the degrees of  and  as
follows. When degrees are clockwise, they are positive; when degrees are anticlockwise,
they are negative. The scale of both  and  should be (-π, π].
7
Gamma Degree Analysis
Cos  histogram
In one dimension space
Step=10
Step=50
Step=200
Step=1000
Step=5000
Step=10000
Figure 11. Cos  of Random walk in one dimension space with different steps.
In two dimension space
Step=10
Step=50
Step=200
Step=1000
Step=5000
Step=10000
Figure 12. Cos  of Random walk in two dimension space with different steps.
8
In a circular orbital with radius of 3
Step=10
Step=50
Step=200
Step=1000
Step=5000
Step=10000
Figure 13. Cos  of Random walk in one dimension space with different steps.
Conclusion:
For random walk in the one dimension space, particle just can move forward and
backward,  should be zero or pi. There are equal probabilities for both zero and pi, which
agrees the Gaussian distribution of random data.
Compared with random walk in two dimension space, the histograms of cos  values
for random walk in restricted circular orbital have less density distribution near zero. From
the picture of cosine function shown as follows, we know the probabilities during the two
noted regions, near /2 and -/2, decrease in random walk in circular orbital.
9
Sin  histogram
In one dimension space
Step=10
Step=50
Step=200
Step=1000
Step=5000
Step=10000
Figure 14. Sin  of Random walk in one dimension space with different steps.
In two dimension space
Step=10
Step=50
Step=200
Step=1000
Step=5000
Step=10000
Figure 15. Sin  of Random walk in two dimension space with different steps.
In a circular orbital with radius of 3
10
Step=10
Step=50
Step=200
Step=1000
Step=5000
Step=10000
Figure 16. Sin  of Random walk in circular orbital with different steps.
Conclusion:
For random walk in one dimension space, the histograms of sin  show two almost
equal distribution regions. Because  should be zero or , sin  should be close to zero. The
two regions may be relative to  values equal to zero or , respectively.
The histograms of sin  values for random walk in restricted circular orbital have
higher density distribution with positive values of degrees near zero and  than those for
random walk in two dimension space. As shown in the sin picture, the noted regions have
higher probabilities.
4
3
2
1
Now, we know the probabilities of regions of 1 and 3 increase, while the probabilities of
ones of 2 and 4 decrease.
I have no ideas how to explain these more now. I will continue analyze more them till
understanding the physical meaning.
11
Table 1.  degree distribution of random walk
Sample
In one dimension space
In two dimension space
In circular orbital
I
II
0.503
0.249
0.360
0
0.250
0.133
Percent (%)
III
IV
0
0.251
0.136
0.497
0.251
0.371
0
π
0.503
0
0
0.497
0
0
NOTE: Percent data are calculated from the simulation with step numbers higher than
1000.
:
II
III
I
IV
Region I: [-π/4, π/4]
Region II: (π/4, π/2) & [-π/2, -π/4)
Region III: [π/2, 3π/4) & (-3π/4, -π/2)
Region IV: [-3π/4, -π) & [3π/4, π]
II
III
Conclusion:
Random walk along the one dimension has only 0 and π, which average the
probability of 1. In the two dimension space,  is relative to the degree of the next motion.
It should be equal at any direction for random motion. Thus, the four regions of random
walk in two dimension space have the equal probabilities, about 0.25 per each. For random
walk in circular orbital, the probabilities focus more on regions of both I and IV. This
conclusion has been shown above, the regions of 1 and 3 in picture below from the cos and
sin histogram analysis.
4
3
2
1
12
Theta Degree Analysis
Cos  histogram
In one dimension space
Step=10
Step=50
Step=200
Step=1000
Step=5000
Step=10000
Figure 17. Cos  of Random walk in one dimension space with different steps.
In two dimension space
Step=10
Step=50
Step=200
Step=1000
Step=5000
Step=10000
Figure 18. Cos  of Random walk in two dimension space with different steps.
13
In a circular orbital with radius of 3
Step=10
Step=50
Step=200
Step=1000
Step=5000
Step=10000
Figure 19. Cos  of Random walk in circular orbital with different steps.
Conclusion:
For random walk in the one dimension space, cos  distributes mainly near 1, which is
different from the distribution of cos . (Calculation error?)
Compared with random walk in two dimension space, the histograms of cos  values
for random walk in restricted circular orbital have less density distribution near zero. This
is similar to the distribution of cos .
14
Sin  histogram
In one dimension space
Step=10
Step=50
Step=200
Step=1000
Step=5000
Step=10000
Figure 20. Sin  of Random walk in one dimension space with different steps.
In two dimension space
Step=10
Step=50
Step=200
Step=1000
Step=5000
Step=10000
Figure 21. Sin  of Random walk in two dimension space with different steps.
15
In a circular orbital with radius of 3
Step=10
Step=50
Step=200
Step=1000
Step=5000
Step=10000
Figure 22. Sin  of Random walk in circular space with different steps.
Conclusion:
Here, there are two unsymmetrical distribution regions near zero for sin  histograms.
I will explain it is reasonable below.
For random walk in both two dimension space and circular orbital, the histograms are
also unsymmetrical. There are more distributions in the negative regions, indicating that 
degrees have probabilities to be anticlockwise. Besides, the probabilities near -1 for
random walk in circular orbital are lower than those in two dimension space. Strangely,
from the table 2 (shown below), there are higher percent in region IV in circular orbital
than those in two dimension space. That is, in the scale of (-π, -3* π/4) and (3* π/4, π/4], the
probabilities for random walk in circular orbital are higher those in two dimension space;
however, close to –π, the probabilities for random walk in circular orbital is lower. Maybe
these can be explain from the detailed motion along the different spaces.
Degree  maybe can supply some other information, which is different from those of
degree.
16
Table 2.  degree distribution of random walk
Sample
In one dimension space
In two dimension space
In circular orbital
I
II
0.752
0.345
0.444
0
0.154
0.060
Percent (%)
III
IV
0
0.155
0.058
0.248
0.345
0.438
0
π
0.752
0
0
0.248
0
0
The percents for random walk in one dimension space look strange, but are still reasonable.
They do not violate the process of producing the random data relative to Gaussian
distribution. The percent in the region I, which is higher than IV, can occur at such
situation.
Here,
1
2
3
1
3
2
 is 0, if the direction of third motion
is the same as the foregoing two
motions
 is , if the direction of third motion
is opposite to the foregoing two
motions
At such situation,
moving forward,  is always 0.
=0
Process 1:
Process 2:
=0
=0
= 
Process 3:
=0
Process 4:
=0
=0
moving forward,  is always 0.
`
At this point, move
back, thus, next
motion will make  = 
`
`
`
`
= 
=0
=0
moving forward,  is always 0.
the percent for  = 0 should be higher than  = π.
These above is what I have analyzed. I will check these analyses again. Make sure they
are right. Then, I will analyze the experimental data, and hope these can help us to
analyze our experimental data.
17