If you wished to solve the time dependent case of the Collisionless Boltzmann Equation using finite difference methods, an impractically large grid would be required. As a result, an N-Body simulation uses Monte Carlo methods to solve this equation. By increasing the value of N, greater levels of accuracy may be obtained.
As an example, consider the following example (Barnes, 1996) : we wish
to obtain an estimate for the value of 
. Draw a square of area A
with a circle of area 
inscribed within it, where
. Randomly scatter n points within the boundary of
the square and count the number 
which fall within the area of
the circle. Since the points were scattered randomly, it follows that
the number of points found within a particular area is proportional to
the area itself. Thus by taking the ratio 
and noting
that this is the ratio
, we can estimate 
with a fractional uncertainty of
.
Although the Monte Carlo method isn't a particularly efficient method
for calculating , it does outperform other methods when
evaluating multidimensional integrals. This is because the error
depends only on the number of points and not on the number of dimensions.