Many physical processes such as Brownian motion, electron transport through metals, and round off errors on computers are modeled as a random walk. In this model, many steps are taken with the direction of each step independent of the direction of the previous one. For our model, we start at the origin and take steps of lengths (not coordinates) in the x and y directions of
where there are a total of N steps. The distance from the starting point R is related to these steps by
Now while (2) is quite general for any walk you may take, if it is a random walk then you are equally likely to move forwards as backwards in each step - as well as to the right or left. So on the average, for a large number of steps, all the cross terms in (2) will vanish and we are left with
where 
is the square root of the average squared step size or
root mean squared step size. Note, the same result obtains for a
three dimensional walk. According to (3), even though the
total distance walked is 
, on the average, the distance
from the starting point is only 
.
Here are different methods to generate 2-D unit steps.
