Title: Simulate
the path (Random Walk) of a drunkard.
Basic
Theory:
In this lab, we try to
simulate the motion of a drunkard in two dimensions, which depicts Random Walk.
A drunkard can move in any direction without proper thought and we try to simulate
this process using a random function which determines the next step of the
drunkard. It is an example of Markov Chain because the prior motion of the
drunkard does not influence its current motion.
Procedure:
For simulation of
drunkard’s random walk, we first specify an initial position with randomly
distributed values of motion in X and Y directions (dx and dy). These will
characterize that the drunkard moves randomly in a twodimensional space. This
can be described using a random function in Excel as,
dx
=12*RAND()
dx
=12*RAND()
The next values of X and
Y are calculated as
X_{i} = X_{i1}
+ dx_{i}
Y_{i} = Y_{i1}
+ dy_{i}
This formula shows that
the points will be plotted with respect to their last values, and the values
prior to that do not make direct contributions. Hence, we can observe the
Markov property in Brownian motion.
After that, we generate a
Scatter plot of the X and Y values and see the drunkard’s motion. We can
experiment with values of dx and dy to see the changes in the behavior of
particle motion.
Thus, we were able to
visualize a drunkard’s random walk using random distribution and assuming
Markov property.
Sample
Data:
Table1: Sample Data for drunkard’s
motion is
dx

dy

X

Y

0.838546

0.32853

0

0

0.673706

0.94962

0.673706

0.94962

0.59668

0.811554

0.077027

0.13807

0.59386

0.346488

0.51683

0.208419

0.78065

0.295288

0.26382

0.503707

0.16151

0.428357

0.102307

0.932064

0.11278

0.908384

0.01047

1.840448

0.17453

0.235414

0.18501

2.075862

0.76316

0.10395

0.94817

1.971913

0.44464

0.53974

1.3928

1.432171

Output:
Below are the graph plots
for drunkard’s motion up to 300 steps:
Conclusion:
Hence, we could see that
the Drunkard’s motion was consistent with the random values of dx and dy. The
more the values of dx and dy deviate in each new step of the drunkard, the
obtained curve gets more zigzag.