2004 Burt, Scott Russell by wfq74180


									                                      Burt, Scott
            Understanding Isolated Systems of Hard Spheres
                    Faculty Mentor: Randy Shirts, Chemistry and Biochemistry

Chemical modeling is an increasingly important tool in modern research. Despite vast
improvements in computational speed, most systems of interest are so complex that simulations
must be carried out on model systems that are much smaller and simpler than the system of
interest. However, such simplified systems often contain constraints that do not apply to the
larger system. This causes significant deviations from the true behavior of the corresponding
macroscopic system. Such deviations are known as finite size effects. Understanding these
effects are important for correctly extrapolating simulation results to the macroscopic system of
interest. For example, chemical simulations are often performed under conditions of constant
total energy (i.e. an isolated system), which makes understanding finite size effects in isolated
systems important.

Several years ago, Dr. Shirts created the Boltzmann1 program to help undergraduates better
visualize molecular motion. This program simulated the behavior of gas molecules in a box. Its
purpose was to demonstrate how, over time, the values produced by this simulation approached
the values predicted by the Boltzmann energy and velocity distribution. However, it soon
became apparent that for small systems, there are significant deviations from the predicted
results. Dr. Shirts recognized these deviations as finite size effects and subsequently calculated
the expected energy and velocity distributions for finite, isolated systems.

In 2002, I designed and wrote a new computer program to simulate the motion of hard spheres in
1, 2 and 3 dimensions with both reflecting and periodic boundary conditions while gathering a
variety of statistical information about the system. This software allowed me to confirm Dr.
Shirts’ previous results as well as to begin exploring other areas where finite size effects might
be important. During 2003 we used my software to explore the finite size effects in the mean
free path and collision lifetime distributions as well as corrections to the virial coefficients. As
expected, we observed deviations from the macroscopic distributions, however, the origins of
these finite size effects were not immediately apparent.

During winter semester of 2004, I helped Dr. Shirts derive the viral coefficients from first
principles in order to explain the observed finite size effects. The virial equation of state gives a
more accurate description of a gas than is possible with the ideal gas equation: PV/nRT = 1 +
B2ρ + B3ρ2 + … Where B2, B3, … are the 2nd, 3rd, … virial coefficients. Within the framework
of statistical mechanics, the pressure is easily obtained from the partition function, which in turn
is related to a configuration integral. The details are beyond the scope of this report, but for the
case of hard spheres, the configuration integral reduces to an ND dimensional integral over all
the allowed positions of the spheres. The difficulty lies is determining the boundary conditions
and then performing the integration exactly.

I was responsible for solving the boundary conditions for systems of 2 and 3 spheres in periodic
boundaries as well as finding analytic solutions to some of the difficult integrals that arose from
these boundary conditions. By the end of April 2004, Dr. Shirts, Aaron Johnson and I were
successful in solving the integrals for systems of 1 and 2 spheres with both periodic and
reflecting boundaries in 1, 2 and 3 dimensions. We have solved much of the configuration
integrals for systems of 3 spheres, but there are still some problems with the boundary conditions
for 3-body overlap as well as some difficult integrals that have yet to be solved. Our results
reproduce many of the finite size effects that we observed in simulations, but there are still some
unexplained effects that Dr. Shirts is continuing to pursue. Dr. Shirts and I presented this
research in several poster sessions during the 2004 American Chemical Society (ACS) national
conference in Anaheim, CA. We have also submitted several papers to be published at the
March 2005 ACS conference.

My final project concerning isolated systems of hard spheres was creating a new version of the
Boltzmann1 program. The center for instructional design at BYU had attempted, unsuccessfully,
for three years to create a replacement for Dr. Shirts’ original Boltzmann program. Over the
summer of 2004 Ben Lemmon and I were able to design and create a replacement2 for this
software. The new version simulates the motion of hard spheres in 1,2 or 3 dimensions with
reflecting or periodic boundary conditions. The graphical interface displays the motion of the
spheres and also allows a user to control the conditions of the simulation and observe how the
system behaves over time. Furthermore, the interface is designed to show the user how the
statistics of the simulation compare to the statistics of the macroscopic system (e.g. Maxwell-
Boltzmann energy and velocity distributions) as well as the exact distributions for isolated, finite
systems. The core of this software is based on the program that I wrote for Dr. Shirts in 2002
and thus can be used both as a research quality simulator to study finite size effects as well as a
visual aid to help students understand an idealized fluid. Ben Lemmon and I wrote the program
in Java to allow it to be platform independent and easily distributable. The program is available
for download free of charge and is currently being used in most freshman chemistry courses at
BYU as well as in universities across the United States and in several foreign countries. This
software has also been submitted for publication in media supplements for chemistry textbooks.

Figure 1. Sample screenshot from the new Boltzmann 3D program

1. Shirts, R. B. (1995). Boltzmann, A Kinetic Molecular Theory Demonstrator. Trinity Software.
2. Shirts, R. B. (2004). Boltzmann 3D. http://chemwww.byu.edu/faculty/rbs/RESEARCH/

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