Variational Principles in Conformation Dynamics - FU Berlin, FB MI

4 I. IntroductionFigure I.1.: An internal coordinate of a small molecule transitioning between two differentconformations, [Weber, 2010, ch.1.1].can be related to average waiting times, the quantities of interest. Therefore, estimatingeigenvalues and eigenfunctions of the propagator is an interesting task. Markov state models,[Prinz et al, 2011], have been frequently used to this end. The main idea is to discretizethe state space into a finite number of sets, and then approximate the true propagator byafinite-dimensionalmatrixoperator. Theeigenvaluesofthismatrixcanbeusedasanapproximationof the real eigenvalues. However, the quality of such an approximation largelydepends on the choice of discretization. Finding a good discretization requires the use ofclustering techniques to group the data. In the following, we present a different approach toapproximating the dominant eigenvalues and eigenfunctions, not based on partition of unitymembership functions, but on the use of smooth functions in combination with a variationalprinciple. The main idea is to use the shape of the molecular energy function to choose thebasis functions needed for the approximation. We hope that this can be done without theapplication of a clustering method to sampled data and thus help to avoid many numericalinstabilities. We will develop the theory in chapter II, thenapplythemethodtoonedimensionalexamples of a diffusion process in chapter III, andfinallytackleahigherdimensionalproblem in chapter IV. In the end, we hope that this might lead to a robust and computationallyaffordable method to compute eigenvalues and relevant time scales of stochasticprocesses stemming from real world examples.