Scott Burt and Dr. 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.
References
- Shirts, R. B. (1995). Boltzmann, A Kinetic Molecular Theory Demonstrator. Trinity Software.
- Shirts, R. B. (2004). Boltzmann 3D. http://chemwww.byu.edu/faculty/rbs/RESEARCH/Boltz3Dhome.html