David Aristoff, CSU Math
Dimension reduction in molecular dynamics
The Mori-Zwanzig formalism is a very general technique for reducing the dimension of systems of ODEs. In the applications we have in mind, the ODEs are given by Hamiltonian dynamics. The formalism allows one to write down an equation for "coarse grained variables" associated with a projection map. The coarse grained variables can be, for example, a subset of the original variables, or the nodes associated with a finite element projection. Such coarsening techniques are extremely valuable, as direct simulation of Hamiltonian dynamics is not practical for high dimensional systems. Of course, the dimension reduction comes at a cost: the equation for the coarse variables contains the integral of a "memory kernel." This kernel depends on the dynamics orthogonal to the projection, and is seemingly impossible to estimate in general. We discuss specific examples in which the memory kernel should be controllable.