Theoretical foundations of dynamical Monte Carlo simulations

Monte Carlo methods are utilized as computational tools in many areas of chemical physics. In this paper, we present the theoretical basis for a dynamical Monte Carlo method in terms of the theory of Poisson processes. We show that if: ( 1) a “dynamical hierarchy” of transition probabilities is created which also satisfy the detailed-balance criterion; (2) time increments upon successful events are calculated appropriately; and (3) the effective independence of various events comprising the system can be achieved, then Monte Carlo methods may be utilized to simulate the Poisson process and both static and dynamic properties of model Hamiltonian systems may be obtained and interpreted consistently.