UC Santa Cruz

The Mori-Zwanzig (MZ) formulation is a technique from irreversible statistical mechanics that allows the development of formally exact evolution equation for the quantities of interest such as macroscopic observables in high-dimensional dynamical systems. Although being widely used in physics and applied mathematics as a tool of dimension reduction, the analytical properties of the equation are still unknown, which makes the quantification and approximation of the MZ equation arduous tasks. In this dissertation, we address this problem from both theoretical and computational points of view. For the first time, we study the MZ equation, especially the memory integral term, using the theory of strongly continuous semigroups, and establish an estimation theory which works for classical and stochastic dynamical systems. In particular, some recent results from the H\"ormader analysis of hypoelliptic equations are applied to get exponential decay estimates of the MZ memory kernel. We also develop a series expansion technique to approximate the MZ equation, and provide associated combinatorial algorithms to calculate the expansion coefficients from first principles. The new approximation methods are tested on various linear and nonlinear dynamical systems, with convergence results obtained both theoretically and numerically. Further developments of the Mori-Zwanzig formulation based on the mathematical framework provided in this work can be expected, which can be used in general dimension reduction problems from physics and mathematics.