Averaging for Diffusive Fast-Slow Systems with Metastability in the Fast Variable

Abstract

The thesis is based on conditional averaging of fast degrees of freedom (DOF). To this end, one considers gradient systems with different time scales: slow DOFs x that vary on a time scale of O(1) and fast DOFS y on a time scale of O(ε). Under certain conditions the fast variables can be eliminated from the original equation of motion by averaging according to the probability distribution corresponding to the exploration of the accessible fast state space. However, if the fast state space exhibit metatstable subsets, that is, subsets from which the fast motion will most probably exit only on some scale of order O(1) or even larger, the standard averagig scheme may fail to reproduce the effective dynamics of the original system. This problem can be solved by restricting the averaging procedure to the metastable subsets of the fast state space, respectively, and coupling the resulting averaged equations of motion in the slow variable x by means of a transition process; the associated rate matrix is then obtained by means of the expected exit times of the fast process from the metastable sets for fixed slow variable x. The thesis consists of two fundamentally different approaches to retain the principle of 'Conditional Averaging'. The first approach (Chapter 2) is dedicated to obtain a deeper insight into the nature of the conditionally averaged system. To this end, we take advantage of the methodology employed to extract the effective dynamics in x. The result tells us that each metastable subset of the fast dynamics is connected to one averaged equation. This motivates the idea to construct a system of fast-slow equations which allows for the incorporation of temporal fast scale effects in a natural way: the fast motion within one metastable subset is approximated by an irreducible Ornstein-Uhlenbeck subprocess that corresponds to a stochastic differential equation. We then parameterize a Markov chain model that controls the switches from one (sub)process to the other according to the transition rates between the metastable subsets of the original dynamics. A reduced system in the slow variable is then obtained by applying the well-known averaging results to each of these stochastic differential equations. For the second approach (Chapter 3), we switch attention to the formal analysis of the Fokker-Planck equation in the asymptotic limit ε to zero, and reconsider the derivation of the effective x dynamics over order unity time scale. Separating the time scale over which the x dynamics proceed and the time scale of the metastable transitions will enable us to obtain a categorization of the various kinds of long-time effective behaviour that can emerge from the system. The conditionally averaged system is then retained as one of the different scenarios that now include metastable transitions that can also occur along the dynamics of the slow variable x.