3.6
 Proving Average Squared Difference (Optional)
What
is the value of the average square displacement of a random walker
after N steps? Here we show two ways to calculate this average square
displacement.
First
Method: Average Square Displacement from Pascal's Triangle
Look at Pascal's
Triangle, Figure 3.7.
Pick
out the row of circles representing displacements after two steps.
A total of 4 paths are available for entering this row (the denominator
of the average in the equation below). Two paths lead to zero displacement,
one path leads to a displacement +2, and one path leads to a displacement
2. Each of these paths is equally likely. In taking the average
of the squares (numerator of the equation below), there is one entry
for (2)^{2} = 4, two entries for (0)^{2} = 0, and
one entry for (+2)^{2}. Therefore the average of the square
for two steps is:

1(2)^{2}+2(0)^{2}+1(+2)^{2}
4

= 
8
4

= 2. 

Now calculate
the average of the squares of the displacements after three steps.
A total of 8 paths are available for entering this row. Three paths
lead to +1 final displacement. Three paths lead to 1 final displacement.
One path leads to +3 and one path to 3 displacement. Follow the same
steps as above to calculate the average of the squares of the displacements
after three steps:

1(3)^{2}+3(1)^{2}+3(+1)^{2}+1(+3)^{2}
8

= 
24
8

= 3. 

Show that
for the row representing displacement after four steps the average
of the squares of the displacements is 4. What is the average of the
squares of the displacements after five steps?
Do
you see a pattern? The average of the squares of the final displacements
is equal to the number of steps. A graph of the ideal mean square
displacement vs. the number of steps is simply a straight line.
Second
method: Calculating the Average Square Displacement using Algebra
Notation:
Scientists often use the symbol x to represent displacement
along a line, x^{2} to represent the square of this displacement,
a subscript N to represent "after N steps,'' and a bracket
á ñ to represent average value. Then our argument from Pascal's
triangle is that:
(average of the squared displacements after N steps) =
number of steps N, 

which
is written
(x_{N})^{2} = N. 

We already
obtained this result using Pascal's Triangle. Next we calculate the
same answer using algebra.
Suppose
the walker has taken n steps and is now at position x_{n}.
What do we expect the value of (x_{n})^{2} to be? Start by
asking where the walker will be at the next step n+1. If we know
where the walker is now (i.e., x_{n}) then after the next
step the walker can be a step to the right or a step to the left;
either at
x_{n+1} = x_{n}
+ 1 (step right) 

(1) 
or at
x_{n+1} = x_{n}
 1 (step left). 

(2) 
Which
of these will it be? We cannot say for sure. In a random walk both
are equally likely. So we take an average: Square both sides of Eqs. (1)
and (2) and take the average of the two. Again, use the bracket to mean
average value. Then
(x_{n+1})^{2} = 
(x_{n}+1)^{2}+(x_{n}1)^{2}
2

= 
x_{n}^{2}+2x_{n}+1+x_{n}^{2}2x_{n}+1
2

, 

or
(x_{n+1})^{2} = 
2(x_{n})^{2}+2
2

, 

or
(x_{n+1})^{2} = (x_{n})^{2}+1. 

(3) 
What does this equation
mean? Start with n = 0, the zeroth step (or no step at all). Then (x_{0})^{2} = 0. For the first
step, Eq. 3 tells us that (x_{1})^{2} = (x_{0})^{2}+1 = 0+1 = 1, which
we knew already without doing this calculation. From this, it follows that (x_{2})^{2} = (x_{1})^{2}+1 = 1+1 = 2 and
(x_{3})^{2} = 3 and, in general
(x_{N})^{2} = N. The result?
The average squared displacement after N steps is simply N:
(x_{N})^{2} = N. 

This is
the same result we obtained from studying Pascal's Triangle.
What we stated
above is an ideal average, i.e., an average we would also obtain over an
infinite number of trials. For a finite number of trials, for example the
average of 250 walkers, this is our "best guess'' of the final squared
displacement after N random steps along a line. In general, observed values
approach the "best guess'' for a very large number of trials.
Previous: SimuLab
8  Measures of Average Squared Displacement