Coarse-grained stochastic processes and Monte Carlo simulations in lattice systems
Diverse scientific disciplines ranging from materials science to catalysis to biomolecular dynamics to climate modelling involve nonlinear interactions across a large range of physically significant length scales. In this talk we present a new class of coarse-grained stochastic processes and corresponding Monte Carlo simulation methods describing computationally feasible mesoscopic length scales, derived directly from microscopic lattice systems. Our main paradigm is a microscopic spin flip model for the adsorption and desorption of molecules between a surface and the overlying gas phase, while such types of microscopic, spin flip processes have also been proposed recently as providing prototype models for unresolved features of moist atmospheric convection. Furthermore we discuss spin exchange models for surface diffusion and pattern formation in systems with competing microscopic mechanisms.
We demonstrate analytically and numerically that the new coarse-grained stochastic models can capture large scale structures while retaining significant microscopic information. The requirement of detailed balance is utilized as a systematic design principle to guarantee correct noise fluctuations for the coarse-grained model. The coarse-grained stochastic algorithms provide large computational savings without increasing programming complexity or CPU time per executed event compared to microscopic Monte Carlo simulations. This is joint work with A. J. Majda (Courant Institute) and D. G. Vlachos (Chemical Engineering, University of Delaware).