Mathematical models of complex physical processes often involve large number of degrees of freedom as well as events occurring on different time scales. Therefore, direct simulations based on these models face tremendous challenge. This focus of this talk is on the Mori-Zwanzig (MZ) projection formalism, which has re-emerged recently as a power tool for reducing the dimension of a complex dynamical system. The goal is to mathematically derive a reduced model with much fewer variables, while still able to capture the essential properties of the system. In many cases, this formalism also eliminates fast modes and makes it possible to explore events over longer time scales. The motivation for this work is from molecular dynamics models of material science problems, where only a small fraction of the atomic degrees of freedom are directly responsible for the defect formation and migration. But the methodology has been applied to macromolecular systems as well.
The models that are directly derived from the MZ projection are typically too abstract to be practically implemented. We will first discuss cases where the model can be simplified to generalized Langevin equations (GLE). Furthermore, we introduce systematic numerical approximations to the GLE, in which the fluctuation-dissipation theorem (FDT) is automatically satisfied. More importantly, these approximations lead to a hierarchy of reduced models with increasing accuracy, which would also be useful for an adaptive model refinement (AMR).
Examples, including the nonlinear Schrodinger equation, molecular dynamics models of materials defects, nanoscale heat conduction and molecular models of proteins, will be presented to illustrate the applications of the methods.