To get a feel for how the science and computation support strengthen
each other, we look at some examples. In Figure 
we see
some actual 2D random walk simulations from the Monte Carlo
project. By uncovering correlations in pseudorandom number, the
students have already seen that the random number generators are not
truly random and so know that Monte Carlo simulations may be
suspect. Although random processes are discussed in many undergraduate
science and engineering courses, here the students themselves generate
a random walk and for the first time actually ``see'' what one looks
like. They often ask me if their results are correct--as if I have
seen molecules collide. Given that opening, I tell them that if their
computer simulation makes predictions which agree with experiment,
then it is likely that their is some truth in the simulation. The
simulation is successful if it lets students ``see'' a process which
previously could only be imagined or read about in a book, gives them
a deeper insight into how scientific modeling works, or gives them a
new view of reality. The simulation is truly successful when these
insights cause the student to experience the addicting excitement of
computational science.
To check the validity of models of random processes, in
Figure we give a student's result for the distance
covered after N random steps versus 
. We see that although
the theory of random processes predicts 
for large N, this
holds only on the average after many trials, and even then only if
particular care is used in generating the random walk.
In Figure we present the results of a Monte-Carlo
simulation of radioactive decay. The simulation is based on the
empirical observation that the probability of there being a
spontaneous decay of an excited atom is proportional to the number of
atoms present N and the time period 
of the
observation. If we assume the number of decays 
is
proportional to the probability of decay (in a statistical sense),
then
This leads to the basic algorithm
The results In Figure of the simulation of this
equation show the student that if a meaningful decay rate (i.e. slope
of the curve) can be defined, then it is independent of the initial
number of nuclei decaying. Yet of even more interest, the student sees
that for large numbers of atoms the decay can be approximated well by
an exponential (a straight line on this semilog plot), but that for
small N the process is stochastic in nature. Furthermore, this graph
shatters the student's prejudice that the analytic result (exponential decay) is
``exact'' while the numerical result is an approximation; here the numerical
simulation holds for all values of
N while exponential decay clearly becomes a poorer and poorer
approximation as N decreases (which of course is what happens in
nature).
Although we cannot present it here, the student's belief in the model
and understanding of nature is further strengthen by the sonification
work of Hans Kowallik[5]. He has created a virtual
Geiger-counter web tutorial which converts the decay rate into a
series of 1's and 0's, opens up a sound player, and then plays the
decay simulation. It's rather satisfying to hear your simulation
actually sounding like a Geiger counter and to know that it all follows
from an idea as simple as Equation ().
My final example deals with the numerical solution of the ordinary differential equation resulting from applying Newton's second law of motion to a particle bound by a potential which always attracts particles to the origin,
For n=2 this is the familiar harmonic oscillator whose solutions most physics
students study over and over again, and which provides a good test case for the
numerical method. As n increases, the potential looks more and more like a square
well and x(t) looks more and more like a sawtooth function. This is a good mix
of physics and computing since it shows the student that 11-place accuracy is
attainable for 1000's of periods, and that for 
the time dependence of the
potential and kinetic energies are very different (which we hasten to
point out is a consequence of the virial theorem). Finally in
Figure we see the results of a
number of computations in which the period of the oscillation is
plotted as a function of both the amplitude of the oscillation and the
power n in (). We see that for n=2 there is no
amplitude dependence to the period (which is expected theoretically
and which means the algorithm is working), but that for larger values
of n the period decreases as the amplitude is made larger. This is
an exciting result, in part because it is not obvious, in part because
it is for a system not accessible to analytic solution, in part
because it is so easy to obtain numerically, and in large part because
it shows how easy it is to understand systems which were too hard for
previous generations to study.
