Reformulation of the path probability method and its application to crystal growth models

The microscopic master equation of a system is derived within the framework of the path probability method (PPM). Then, by extending Morita's method in equilibrium statistical mechanics, the path probability function constructed microscopically can be systematically decomposed to result in the conventional path probability function of cluster approximation when correlations larger than the chosen basic cluster are neglected. In order to critically compare the master equation method with the PPM, the triangle approximation is treated by both methods for crystal growth models. It is found that the PPM gives physically satisfactory kinetic equations, while the master equation (supplemented with a cluster probability in the superposition approximation) does not. The triangle PPM calculation considerably improves the result of the pair approximation for crystal growth velocity in the solid-on-solid model, and compares well with Monte Carlo results.