Introduction

With the rapid increases of computer power and progresses in software development, computer modeling is playing an ever important role in the technological world covering a wide range of fields, including drug design in biotech, control and signal error corrections in telecommunications (your modem at home that allows you to connect to internet at speed higher than 600 bps, for example, has to do such job to make sure signals don't screw up while bundled together), pattern/voice recognition, failure analysis, and of course, materials design. The essence of computer modeling is to minimize the error subject to constraints, with the error being defined through a set of rules or models. In the field of materials science, this error is the deviation from the minimum free energy (or internal energy at zero temperature) associated to a set of macroscopic thermodynamics parameters (temperature, pressure, number of particles). The modeling can be ab initio, i.e., defined based on quantum mechanical principles, or empirical which are constructed coarse-graining the microscopic degrees of freedom we understood and find irrelevant for our purposes. Two basic modeling tools are molecular dynamics and Monte-Carlo simulations.

Computational Researches at MSC

Here at MSC, Caltech, I am primarily engaged in developing new ab initio density functional based molecular dynamics. The goal is to have a computer software that works for all the elements of the periodic table, that scales quasi linearly with the size of the system so that large scale simulations are possible, and to be compatible with other quantum chemistry methods so that the accuracy of density functional calculations can be controled if needed. This research is now in its final stage. We have already completed the critical part of properly working electronic structure code ( Click here for a publication ). I am currently working on implementing the Generalized Car-Parrinello method for doing ab initio molecular dynamics.

Another computational method that interests me is the coarse graning of atomistic Hamiltonian into macroscopic simulations. This is a very important link that if succeded, will ultimately enables us to predict macroscopic properties from first principles of quantum mechanics. The result will be well controled engineering processes and design. Right now, there is no theory guarantees that such link is possible. That is, it is not yet mathematically clear if a macroscopic Hamiltonian can be constructed from a microscopic one. However, if we are willing to introduce the statistical/thermodynamics into the process of such coarse-graining, then the answer, in my view, has to be positive. The foundation of such assertion? See Feynman's text book. The more subtle thing is, if we introduce thermal fluctuations into the coarse-graining, how do we avoid double counting? For the microscopic degrees of freedom, that is ok, since they do not re-appear in macroscopic simulations. For the macroscopic degrees of freedom, this is more problematic, and I think the no uniqueness comes from here. It is prossible that such problem can be solved with ``minimal action'' kind of method, but I haven't gone through it yet.

The more tedious things is how to proceed. Well, we have to define a set of thermodynamic variables. We consider the small reservior, the one we simulate with microscopic Hamiltonian to be linked with larger reservoir represented with thermodynamic variables. We carried out coarse graining of the equilibrium microscopic Hamiltonian subject to such boundary condition. The resulting Hamiltonian is thus compatible with the larger reservoir. Such process can be done in parallel spatially. The result is a macroscopic simulation with microscopic accuracy subject to a finit set of thermodynamic/boundary constraints.