Title: Simulation
of Brownian motion of smoke particles.
Basic
Theory:
In this lab, we try to
simulate the motion of multiple particles in two dimensions, which depicts
Brownian motion. Brownian motion is a stochastic model in which changes from
one time to the next are random draws from a normal distribution. It is a
physical phenomenon which can be observed, for instance, when a small particle
is immersed in a liquid. The particle will move as though under the influence
of random forces of varying direction and magnitude. There is a mathematical
idealization of this motion that allows us to simulate the successive positions
of a particle undergoing Brownian motion. It is an example of Markov Chain.
Procedure:
For simulation of
Brownian motion, we first specify a collection of random particles with
randomly distributed values of motion in X and Y directions (dx and dy). This
will characterize that the particles move randomly in a twodimensional space. This
can be described using a random function in Excel as,
dx
=12*RAND()
We can use dx =13*RAND()
for a more scattered motion. This is same for y as well.
Now, we assign the drift
values for X and Y points. They will further define the movement of particles
due to some specific drift forces that act on them. So, the positions of
particles are defined by
X_{n}
= X_{n1} + Drift(x) + dx
Y_{n}
= Y_{n1} + Drift(y) + dy
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 particle motion. We can
experiment with drift values to see the changes in the behavior of particle
motion.
Thus, we were able to
visualize Brownian motion using random distribution and assuming Markov
property.
Sample
Data:
Table1: Sample Data for Brownian
motion is
dx

dy

X

Y

0.89132

0.9203

0

0

0.208206

0.7602

1.208206

0.239804

0.021238

0.78757

2.229443

0.452233

0.78564

0.614743

2.443803

2.066976

0.765821

0.259244

4.209624

3.32622

0.2025

0.748862

5.007128

5.075081

0.03483

0.09562

5.972299

6.170702

0.89046

0.70513

6.081844

6.465568

Output:
Below are the graph plots
for Brownian motion for 300 particles:
Conclusion:
Hence, we could see that
the Brownian motion of particles was consistent with chosen drift values. The
more the drift in any one direction, the obtained curve gets wavier. We can further
increase the precision of the simulation by increasing the number of random points,
thus being able to observe larger number of particles. However, the excel worksheet
will become too slow if we generate a lot of random numbers because of large
number of calculations.
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