Faster Monte Carlo simulations

For Monte Carlo simulations of systems of size M, either kinetic simulations or equilibrium simulations that use the method of Bortz, Kalos, and Liebowitz [J. Comput. Phys. 17, 10 (1975)], the best computer time per event has been O(${\mathit{M}}^{1/2}$). We present two methods whose computer time per event is O(${\mathit{M}}^{1/\mathit{K}}$) or O(logM). In practice, for typical simulation sizes, K = 4 or K = 5 is fastest, requiring even less computer time than the O(logM) method. For typical simulation sizes, we are able to achieve speedup factors of 5 to 7 over the O(${\mathit{M}}^{1/2}$) technique.