This talk reviews the basic computational techniques for carrying out multi-scale simulations using statistical methods, with the focus on simulations of epitaxial growth of semiconductor nanostructures. First, the statistical-physics background behind Monte Carlo simulations is briefly described. The kinetic Monte Carlo (kMC) method is introduced as an extension of the more wide-spread thermodynamic Monte Carlo methods, and algorithms for kMC simulations are discussed in some detail. The step from the atomistic picture to the more coarse-grained description of Monte Carlo simulations is exemplified for the case of surface diffusion. In this context, the aim is the derivation of rate constants from knowledge about the underlying atomic processes. Both the simple approach of Transition State Theory, as well as more recent approaches using accelerated molecular dynamics are reviewed. Finally, I address the issue of simplifying assumptions introduced in practical Monte Carlo simulations in order to reduce the complexity of 'real' molecular processes. Different 'flavors' of kMC simulations and the potential pitfalls related to the reduction of complexity are presented in the context of simulations of epitaxial growth.
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