Class of Monte Carlo algorithms for dynamic problems leads to an adaptive method

Abstract

We introduce a class of Monte Carlo algorithms that solve a dynamic problem defined by the transition rates and the initial state of a discrete system. This class contains the method of Bortz, Kalos, and Lebowitz (BKL) [J. Comp. Phys. 17, 10 (1975)] as a limit. We show that introducing a constant time step in a Metropolis algorithm leads to an approximation of the solution in which the system relaxation times are underestimated. This can be corrected if the time step is an adequate stochastic variable. Thus, we are able to define kinetic Metropolis algorithms and generalize them in a case of nonconstant numbers of attempt configurations. The algorithm class allows us to introduce a useful method in which the calculation of transition rates are exploited for the next step in an adaptive way. This method corresponds to a kinetic Metropolis algorithm when the rejection probability is reasonable and becomes similar to the BKL method otherwise. We describe and compare four different algorithms applied to a physical example about the diffusion in lattice gases.