Extracting Macroscopic Stochastic Dynamics: Model Problems

Abstract

The purpose of this work is to shed light on an algorithm designed to extract effective macroscopic models from detailed microscopic simulations. The particular algorithm we study is a recently developed transfer operator approach due to Schütte et al. [20]. The investigations involve the formulation, and subsequent numerical study, of a class of model problems. The model problems are ordinary differential equations constructed to have the property that, when projected onto a low-dimensional subspace, the dynamics is approximately that of a stochastic differential equation exhibiting a finite-state-space Markov chain structure. The numerical studies show that the transfer operator approach can accurately extract finite-state Markov chain behavior embedded within high-dimensional ordinary differential equations. In so doing the studies lend considerable weight to existing applications of the algorithm to the complex systems arising in applications such as molecular dynamics. The algorithm is predicated on the assumption of Markovian input data; further numerical studies probe the role of memory effects. Although preliminary, these studies of memory indicate interesting avenues for further development of the transfer operator methodology.