ITR: Mesoscopic Modeling and Simulation: A Novel Approach to Monte Carlo Methods

Proposal # 0219211

PI: Markos A. Katsoulakis

Institution: University of Massachusetts Amherst

Title: ITR: Mesoscopic Modeling and Simulation: A Novel Approach to Monte Carlo

ABSTRACT

This project is concerned with a novel framework for Monte Carlo simulations based on recently developed coarse-grained stochastic mesoscopic models. Mesoscopic models are stochastic partial differential equations which are rigorously derived as asymptotic limits from microscopic Monte Carlo (MC) algorithms by means of techniques from non-equilibrium statistical mechanics. Although such models describe the mesoscopic scales, which are much larger than the underlying molecular scales, they still include detailed microscopic information on particle interactions and dynamics and can systematically model anisotropies and multiple micromechanisms. Another attractive feature of mesoscopic models is the inclusion of random fluctuations derived directly from the underlying master equation and yielding important nucleation and pattern formation and selection mechanisms. Finally, formal and rigorous asymptotics using Large Deviation and WKB expansions as well as preliminary numerical simulations indicate that the MC algorithms and the corresponding stochastic mesoscopic models produce essentially identical results for such delicate quantities as nucleation rates and phase transitions. The main research objectives of this proposal are: (a) to develop non-equilibrium coarse-grained MC algorithms by numerically solving the stochastic mesoscopic equations using highly efficient spectral-based methods and carry out detailed benchmarkings against conventional MC, and (b) to apply the proposed computational tools to applications arising in pattern formation in advanced materials and molecular separation in nanoporous films.

Because of their fundamental nature and their versatility in describing complex out-of-equilibrium interactions between atoms and molecules, molecular dynamics simulations and Monte Carlo (MC) algorithms have become preeminent computational tools for science and engineering research. With the advent of enhanced computing capabilities, these methods can provide unprecedented insights into numerous problems ranging from physicochemical and biological processes to biomaterials, drug design, pattern recognition, and image processing. Despite their widespread use and the substantial progress in related computational methods, molecular algorithms are limited to short length and time scales. Hence, they are capable of simulating only a relatively small number of atoms or molecules for quite short time periods. On the other hand, device sizes and morphological features observed in experiments often involve much larger spatial and/or temporal scales. A major obstacle in meeting this multiscale modeling challenge is the lack of a rigorous mathematical and computational framework providing a direct link from the atomistic scale to the complex mesoscopic and macroscopic phenomena that are the result of the microscopic interactions. In this direction, our work focuses on developing novel stochastic models and algorithms capable of describing much larger length and time scales than conventional MC simulations while still incorporating microscopic details. We intend to apply the computational methods to provide new insights into two engineering problems, which are currently intractable with conventional MC techniques: (1) the study of self-organizing micromechanisms and their role in pattern formation in advanced materials, and (2) the transport and separation of molecules in nanoporous films and membranes.