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.''

 Q3.31: Does AVG. |x| increase with the number of steps?

 Q3.32: Q32: For a given number of steps, does AVG. |x| have a larger value for more walkers?

The average absolute displacement is not the measure chosen by scientists to describe the random walker, because it does not give the simplest result, as will be shown below. The result is simpler if we take the square of each displacement and then average these squares. The square of a number is positive, even when the number itself is negative. (If the number is zero, its square is also zero.) Therefore the average of squares of final displacement will never be negative. This average is called the average squared displacement or mean square displacement.

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:

• 2 ants had a squared displacement of (-3)2 or 9

• 9 ants had a squared displacement of (-1)2 or 1

• 7 ants had a squared displacement of (1)2 or 1

• 2 ants had a squared displacement of (3)2 or 9

Averaging we get
 2(9)+9(1)+7(1)+2(9) 20 = 2.6
for the average squared displacement. (The average number of steps does not have to be an integer; if one takes one step and another takes two steps, the average is 1.5 steps.) This result will naturally be a bit different for each trial.

Return to the original picture of the wandering ant (Figure 3.4).

1. Start with the ant in the center.

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 x2 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).

Table 3.2: Computing average of the squared displacement of the random walker.
 Walker One Walker Two Walker Three Step x = x2 = x = x2 = Avg. x2 of x = x2 = Avg. x2 of walkers walkers #1 and #2 #1, #2 and #3 1 2 3 4 5

 Q3.33: As more walkers are added, does the graph of the resulting averages approach a pattern? Using yet another color (black?), draw a line that shows this pattern.

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.

1. Bring up the ManyWalkers program and look at the value of "AVG. x2'' at the right of the bar graph.

2. Try different numbers of walkers and different numbers of steps.

 Q3.34: Does AVG. x2 increase with number of walkers, for a given number of steps? In contrast, does AVG. x2 increase with the number of steps, for a given number of walkers?

3. Store the data on the table and examine the Table.

4. Call up the Graph.

 Q3.35: Do you notice anything which might help you predict the value of AVG. x2? In particular, can you predict the value of AVG. x2 for 250 walkers after 30 steps?

 Q3.36: Print out the graph for 250 walkers taking 30 steps. Then run 40 walkers taking 30 steps and print out the resulting graph. Finally, run 10 walkers taking 30 steps and print out the graph. Label these three graphs, lay them side by side, and compare them. What accounts for the differences and similarities between these three graphs?

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
Next: 3.6 - Proving Average Squared Difference