SimuLab 8: Measures of Average Squared Displacement
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
Here are the results of an experiment in which 20 ants each took 3 steps:
|Number of ants||Final displacement|
To find the average square displacement we calculate as follows:
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 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).
|Walker One||Walker Two||Walker Three|
|Step||x =||x2 =||x =||x2 =||Avg. x2 of||x =||x2 =||Avg. x2 of|
|#1 and #2||#1, #2 and #3|
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.
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