Department of Mathematics and Statistics
University of Massachusetts
Amherst, MA 01003
Joint work with Dionisios G Vlachos.
We present a systematic approach to derive mesoscopic theories (i.e. stochastic integrodifferential equations) and macroscopic governing laws of growth velocity and morphological evolution of clusters, obtained directly from microscopic stochastic systems. Examples presented include surface reactions and deposition processes.
We first introduce the microscopic mechanisms at a statistical mechanics level, which describe the particle-by-particle formation and evolution of clusters; these models essentially constitute Monte Carlo algorithms for the phenomena under consideration, and since they are computationally intensive, are suitable only for describing short scales. Often, as is the case in deposition models, the microscopic processes need to be coupled to a continuum mechanics model at a much larger space/time scale. This underscores the need to bridge the discrepancy between the micro- and macro- models by deriving mesoscopic and macroscopic PDE for the evolving clusters, directly from the microscopic particle systems. One of the crucial steps here is the identification of macroscopic quantities (e.g. transport coefficients, surface tension, etc.) in terms of the microscopic parameters, i.e. interaction potentials and type of dynamics
We finally discuss the microscopic validation of the mesoscopic and macroscopic picture through gradient Monte Carlo simulations, and present spectral numerical methods for the derived mesoscopic models.