A systematic approach to the development of fast path integral techniques

As the dimensionality of the physical systems under study continues to grow, it becomes increasingly important that the formal properties of numerical path integral methods be properly characterized. The present dissertation addresses this issue by providing a broad mathematical setting for the development of path integral methods having fast asymptotic convergence and favorable scaling with the dimensionality of the system.; The first Chapter of the thesis provides a mathematical description of the Brownian motion at a level suitable for chemical-physics audience. Most of the properties analyzed are of direct relevance in the context of the Feynman-Kac formula. These properties are then utilized to justify the Feynman-Kac formula in Chapter 2 as well as to design and characterize various path integral techniques in the remainder of the dissertation. In Chapter 3, the direct discretization of the Feynman-Kac formula, the partial averaging method, and the reweighted path integral techniques are discussed. Energy and heat capacity estimators having favorable scaling with the number of path variables, the temperature, and the Monte Carlo techniques are presented together with several numerical examples.; In Chapter 4, a general approach to constructing path integral techniques having fast asymptotic convergence is presented. The method is based on the design of finite-dimensional approximations of the Brownian motion in such a way that the resulting finite-dimensional approximations to the Feynman-Kac formula have arbitrarily large convergence orders. It is shown that the system of functional equations controlling the convergence orders is independent of the potential, hence of the physical model. While the existence of solutions of this system of functional equations is still an open problem, explicit examples having convergence orders 3 and 4 are constructed. The resulting finite-dimensional Feynman-Kac formulas are then used as short-time high-temperature approximations for the Lie-Trotter product rule. A convergence theorem for the Lie-Trotter product rule is justified and utilized to demonstrate the asymptotic convergence of the resulting path integral methods.