Дискуссионный Петрофизический Форум

An overview of spatial microscopic and accelerated kinetic Monte Carlo methods

Journal of Computer-Aided Materials Design
§Є§Щ§Х§С§д§Ц§Э§о Springer Netherlands
ISSN 0928-1045 (Print) 1573-4900 (Online)
§Ї§а§Ю§Ц§в Volume 14, Number 2 / §Є§р§Э§о 2007 §Ф.
DOI 10.1007/s10820-006-9042-9
pp. 253-308
Subject Collection §·§Ъ§Ю§Ъ§с §Ъ §Ю§С§д§Ц§в§Ъ§С§Э§а§У§Ц§Х§Ц§Я§Ъ§Ц

An overview of spatial microscopic and accelerated kinetic Monte Carlo methods
Abhijit Chatterjee1 and Dionisios G. Vlachos1

(1) Department of Chemical Engineering and Center for Catalytic Science and Technology (CCST), University of Delaware, Newark, DE 19716, USA

Received: 6 August 2006 Accepted: 17 October 2006 Published online: 28 February 2007

Abstract The microscopic spatial kinetic Monte Carlo (KMC) method has been employed extensively in materials modeling. In this review paper, we focus on different traditional and multiscale KMC algorithms, challenges associated with their implementation, and methods developed to overcome these challenges. In the first part of the paper, we compare the implementation and computational cost of the null-event and rejection-free microscopic KMC algorithms. A firmer and more general foundation of the null-event KMC algorithm is presented. Statistical equivalence between the null-event and rejection-free KMC algorithms is also demonstrated. Implementation and efficiency of various search and update algorithms, which are at the heart of all spatial KMC simulations, are outlined and compared via numerical examples. In the second half of the paper, we review various spatial and temporal multiscale KMC methods, namely, the coarse-grained Monte Carlo (CGMC), the stochastic singular perturbation approximation, and the ¦У-leap methods, introduced recently to overcome the disparity of length and time scales and the one-at-a time execution of events. The concepts of the CGMC and the ¦У-leap methods, stochastic closures, multigrid methods, error associated with coarse-graining, a posteriori error estimates for generating spatially adaptive coarse-grained lattices, and computational speed-up upon coarse-graining are illustrated through simple examples from crystal growth, defect dynamics, adsorptionЁCdesorption, surface diffusion, and phase transitions.
Keywords Review - Multiscale simulation - Coarse-graining - Mesoscopic modeling - Monte Carlo - Materials - Defects - Diffusion - Crystal growth - Phase transitions - Accelerated algorithms - Binary tree - Efficient update - Efficient search - Tau-leap - Stiff - Stochastic - Computational singular perturbation - Low-dimensional manifold

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Dionisios G. Vlachos
Email: vlachos@udel.edu

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