Normal form transforms separate slow and fast modes in stochastic dynamical systems

Modelling stochastic systems has many important applications. Normal form coordinate transforms are a powerful way to untangle interesting long term macroscale dynamics from detailed microscale dynamics. We explore such coordinate transforms of stochastic differential systems when the dynamics has both slow modes and quickly decaying modes. The thrust is to derive normal forms useful for macroscopic modelling of complex stochastic microscopic systems. Thus we not only must reduce the dimensionality of the dynamics, but also endeavour to separate all slow processes from all fast time processes, both deterministic and stochastic. Quadratic stochastic effects in the fast modes contribute to the drift of the important slow modes. The results will help us accurately model, interpret and simulate multiscale stochastic systems

Atomistic Simulations of Ge on Amorphous Silica Substrates

High-quality Ge substrates have numerous applications, including high-efficiency III-V multijunction solar cells and photodetectors. But the high cost of single-crystalline Ge makes the use of Ge-on-Si virtual substrates more practical for device fabrication. However, the lattice mismatch between Ge and Si leads to a highly strained Ge layer when grown directly on the Si lattice. The high mismatch strain unavoidably leads to defects, primarily dislocations, that degrade the Ge film quality. Several approaches for mitigating these defects have been proposed, including selective epitaxial growth (SEG), in which one employs an amorphous layer (most often SiO2) as a mask to reduce the epitaxial contact between the Ge and Si lattices to lower the mismatch strain. SEG has been demonstrated to successfully produce high-quality Ge films on Si, although defects are not fully eliminated. Further improvements will require quantitative understanding of the underlying atomic-scale mechanisms. In this work, we present a computational framework to atomistically model the components of the SEG system (Si/SiO2/Ge). The model is validated by comparing predictions to experimental observations and ab initio calculations of various properties related to crystalline Si and Ge and amorphous SiO2, as well as combinations of these materials. The framework is then applied to study in detail the deposition of Ge on amorphous SiO2. It is shown that the simulations are able to access experimentally meaningful deposition conditions and reproduce several quantities related to the island size distribution. We then extend our simulation framework for deposition to include coarse projective integration (CPI). CPI is a multiscale modeling technique well-suited for situations, like atomic deposition, in which a system exhibits fast, stochastic processes, superposed onto slowly-evolving dynamics. In particular, we demonstrate an approach for generating atomistic configurations from limited knowledge of an island size distribution, which represents one of the key challenges in applying CPI to atomistic deposition. The results generated here should be easily adaptable to other deposition systems

Variance reduction for the equation-free simulation of multiscale stochastic systems

We study the problem of simulating the slow observable of a multiscale diffusion process. In particular, we extend previous algorithms to the case where the simulation of the different scales cannot be uncoupled and we have no explicit knowledge of the drift or the variance of the multiscale diffusion. This is the case when the simulation data come from a black box "legacy code," or possibly from a. ne scale simulator (e.g., MD, kMC) which we want to effectively model as a diffusion process. We improve the algorithm, using the past simulations as control variates, in order to reduce the variance of the subsequent simulations