By Matchett J.R., et al. (eds.)
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2004). Such instabilities may occur when the time step of the KMC becomes too large to violate the numerical stability criterion of the continuum model, or when rare events happen that create huge variations in the boundary condition or in the source/sink term of the continuum model. This numerical instability is a result of the small size of the KMC simulation box (a problem stemming from our inability to deal with realistically large length scales). Consequently, the KMC response is considerably noisier than what one would have for realistic length scales.
Recently, the mathematical foundations for spatial CG-KMC have been introduced for grand canonical and canonical ensemble simulations of Ising-type systems in Katsoulakis et al. (2003a, b) and Katsoulakis and Vlachos (2003). This work deserves a review of its own. However, in order to put it in context with the other multiscale developments, some exciting developments are briefly summarized. The essence of the method is the creation of a lattice of coarse cells, each consisting of several microscopic cells.
SPATIAL UPSCALING OF DISTRIBUTED (LATTICE) KMC SIMULATION The problem of coarse graining in space is also very important but has received less attention. The overall idea of coarse graining degrees of freedom to move up in scales comes originally from renormalization group theory. An interesting idea revolves around coarse graining of the Hamiltonian using wavelets. , 2005a, b). Coarse graining of the Hamiltonian was also presented by Ishikawa and Ogawa (2002), but it can be shown that the proposed expression does not obey detailed balance.