Kinetic Monte Carlo Method: Mathematical Foundations and Applications for Physics of Low-Dimensional Nanostructures

The kinetic Monte Carlo (kMC) method is an indispensable method for studying atomic and molecular systems, which makes it possible to solve a wide range of problems associated with atomic diffusion, the formation of defects and chemical compounds of various types, as well as the growth and self-organization of nanostructures. In this paper, we consider the fundamentals of the kMC and its modern modifications, both rigid-lattice and off-lattice. Particular attention is focused on constructing self-learning algorithms based on different methods for finding the saddle points of potential energy and on the techniques for the acceleration of the Monte Carlo method. Every considered method is illustrated by relevant examples mostly associated with the physics of metal surfaces.