Kinetic Monte Carlo method: mathematical foundations and applications to physics of low-dimensional nanostructures

The kinetic Monte Carlo method is essential tool for investigation of atomic and molecular systems. It is applicable for the wide range of problems such as the atomic diffusion, the formation of crystal defects and chemical compounds, the growth and the self-organization of nanostructures. In the present review we consider the basic principles of the kinetic Monte Carlo method and its modern modifications both lattice and non-lattice. The special attention is focused on the self-learning algorithms constructed from different saddle point finding methods and the algorithms for kinetic Monte Carlo acceleration. All methods are illustrated by the actual examples, the most of them are connected with the physics of metal surfaces.