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Random movement is important for
understanding complex behavior in everything from atoms and molecules in a
gas to the motion of stars in a galaxy. That’s because – despite
electromagnetic interactions between molecules or gravitational effects
between stars – the large-scale behavior of these objects appear to make
them move randomly. How can we describe the random motion of molecules
in, say, a gas? Molecules are much too small for us to see, so to help
us better understand how the movement of molecules in a gas works we need to
build a model. And, of course, it is best if we start with a simple
example and add complexity until we can develop a more complete
model.
After that, we will examine a
simple two-dimensional model of random motion, and then introduce an
activity allowing you to explore a more complicated version.
Activity #1.
If an ant starts at an anthill
and takes steps of equal length along a line extending away from the
anthill, how far will it be from the anthill after a certain number
(N) steps? Though this seems to be a trivial example, it will allow us
to work through one of the most basic problems in statistical science. We
will choose the direction of the step the ant will take by flipping a coin:
1. If the coin ends up heads, the ant steps right
and x increases by one (x
+ 1).
2. If it is tails, the ant steps left and
x
decreases by one (x
– 1).
A heads or tails is equally likely; therefore it is equally probable that
the ant steps right or left. See the diagram below.
Use a fair coin (something we have discussed already) to represent the
subsequent movement of the ant. To begin, put the “ant” at the
x = 0 position (the center of the
anthill). The ant steps from center position to the next position
x + 1 right or
x – 1
left randomly, depending on whether the coin comes up heads or tails,
respectively.
Procedure:
1. Flip a
coin and move your ant accordingly.
2. After
ten (10) steps, record the final position of the ant.
3. Repeat
ten (10) times.
4. Graph
the trial number on the x-axis vs. the final position on the y-axis.
5. What
is the average final position <
x > of
the ant from all trials?
Activity #2.
Things in nature certainly move in more complicated ways than our ant.
You have probably watched the way a bee flies as it searches for pollen.
The molecules of the air that you breathe move in a similar way. This
type of motion we call a random walk. In this activity, we will take a
random walk.
On a large, open floor, mark a spot on the ground. Again using a fair
coin, stand on the spot and flip the coin. If the coin comes up heads,
turn to the right and take one large step. If the coin comes up tails,
turn to the left and take a large step. Be sure to keep each step
approximately the same length. Do this many times and see where you
end up.
As you try this activity, you will notice that sometimes you go much farther
than you expect and sometimes you end up very close to where you started.
However, if you repeat it many times or get several of your friends to do it
with you with coins of their own, the average distance from the start point
to your final position should produce a nicely predictable relationship.
Procedure:
1. Flip a
coin and move each step accordingly.
2. After
ten (10) steps, record your final position.
3.
Measure the straight-line distance from the start point to your final
position.
4. Repeat
ten (10) times.
5. What
is your average distance <
x > from
all trials?
Extension:
1. Flip a
coin and move each step accordingly.
2. After
five (5) steps, record your final position.
3.
Measure the straight-line distance from the start point to your final
position. Record.
4. Repeat
the experiment but increase the number of trials by five (5). Continue
until you
have completed twenty-five (25) coin tosses.
5. Graph
the number of steps on the x-axis vs. the straight-line distance from
the start
point to your final position on the y-axis.
What is the mathematical
function
described by the graph of steps vs. distance from the start point to your
final position? From this, how far would you expect to be if you
flipped the coin 100 times? A 1000 times? 10000 times? As you
can see, a random walk is not a very fast way to get anywhere!
However, some of the most powerful tools we have in science are the rules of
probability and
statistics, where we
can often predict what will happen on the average even when the process is
completely random.
Activity #3.
Write a simple program that
examines a random walk. One such program is given below (in a language
called
NetLogo™).
Make modifications in the program in order to study the behavior of
ever-increasing random walks.
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to setup
ca
;; clears all inputs
set-default-shape turtles "turtle"
crt 1
;; create one turtle
ask turtles
[ set color
yellow
;; turtle has a pen
set pen-size 1.5
;; big enough to see the line
pd ]
;; and puts it down.
end
to go
ask turtles
[ repeat 10
[ set heading
random 4 * 90
;; set random heading
up/down/left/right
fd 1 wait 0.1 ]]
;; forward a step, wait, and do it 10 times.
end
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This program cycles through a two-dimensional random walk. The
highlighted key parts will be modified later in the Running the model
section. Enjoy!
This is a screenshot of what the program looks like.
Extensions:
1. Change
the number of trials (10, 100, 1000, 10000, etc.).
2. Change
in direction (30º, 45º, 90º, etc.).
3. Change the length of each step by a random amount
(fd 1, 2, 3, 5, etc.). |
