SimuLab 8: Measures of Average Squared Displacement
Return
to the ManyWalkers program. This time pay attention to the
value of "AVG. x'' given at the right of the bar graph.
The symbol x stands for "absolute value of x,'' or "magnitude
of x.''


Here are the results of an experiment in which 20 ants each took 3 steps:
Number of ants  Final displacement 
2   3 
9   1 
7  1 
2  3 
To find the average square displacement we calculate as follows:
Averaging
we get

Return to the original picture of the wandering ant (Figure 3.4).
2. Flip a coin and move the ant one step.
3. Record its position (+1 or 1) in a copy of Table 3.2.
4. Now flip the coin again, move the ant, and record its new position.
5. Continue for a total of five steps, recording the ant's position
after each coin flip.
6. Now square the total distance (displacement) from the starting
point after each coin flip.
7. We want to graph the average squared displacement versus
the number of steps. Plot your data in a distinctive color
on a graph with number of steps along the horizontal axis and
x^{2} along the vertical axis, where x is the displacement.
8. Repeat Steps 1 through 7 using a second ant, again recording
the position after each coin flip.
9. This time take the average of the squared displacements of
the two walkers and plot this in another color (green perhaps)
on the graph.
10. Continue with the third walker, this time taking the average
of the squared displacements of all three walkers after each coin
flip. Plot this in yet another color (maybe blue).
Walker One  Walker Two  Walker Three  
Step  x =  x^{2} =  x =  x^{2} =  Avg. x^{2} of  x =  x^{2} =  Avg. x^{2} of 
walkers  walkers  
#1 and #2  #1, #2 and #3  
1  
2  
3  
4  
5 

Can we make any prediction about the value of the average squared
displacement after many trials? Once more, we can use the computer
to give us many trials.
2. Try different numbers of walkers and different numbers of steps.

4. Call up the Graph.


Predict what you expect the graph to look like when one
walker takes 30 steps. Try it and compare the result with your
prediction.
You can watch the change in the graph as the number of trials increases. To do this, return to the Random Walk program and open the Graph Displacement window.
Previous: 3.5  Measuring Average Distances