Mathematical Strategies for the Coarse-Graining of Microscopic Models

Abstract

Spatial inhomogeneity at some small length scale is the rule rather than the exception in most physicochemical processes ranging from advanced materials’ synthesis, to catalysis, to self-assembly, to atmospheric science, to molecular biology. These inhomogeneities arise from thermal fluctuations and complex interactions between microscopic mechanisms underlying conservation laws. While nanometer inhomogeneity and its corresponding ensemble average behavior can be studied via molecular simulation, such as molecular dynamics (MD) and Monte Carlo (MC) techniques, mesoscale inhomogeneity is beyond the realm of available molecular models and simulations. Mesoscopic inhomogeneities are encountered in self-assembly, pattern formation on surfaces and in solution, standing and traveling waves, as well as in systems exposed to an external field that varies spatially over micrometer to centimeter length scales. It is this class of problems that require “large scale” mesoscopic or coarse-grained molecular models and where the developments described herein are applicable. It is desirable that such mesoscopic or coarse-grained models meet the following needs: They are derived from microscopic ones to retain microscopic mechanisms and interactions and enable a truly first principles multi-scale approach; They reach large length and time scales, which are currently unattainable by micro scopic molecular models;