Monte Carlo as Brownian dynamics

A study is reported of the relationship between Metropolis Monte Carlo (MC), smart Monte Carlo (SMC), and Brownian dynamics (BD) as invented by Ermak. SMC and ED are shown to be formally equivalent in the limit of zero timestep (i.e., Delta t --> 0). However it is not easy to prove the equivalence between MC and ED beyond the trivial zeroth-order term (i.e., not at the order of the mean-square systematic force contribution to the mean-square displacement). Test calculations on model high volume fraction colloidal systems reveal that SMC gives the same dynamics as ED and, in addition, can be employed I,vith larger timesteps than the ED method without any noticeable loss of accuracy or systematic displacement of the averages and time autocorrelation functions (force and sheer stress). (The importance sampling in the SMC method filters out statistically unrepresentative trajectories that lead to algorithmic breakdown with large timesteps in ED.) The gain in efficiency resulting from an increased timestep is reduced somewhat by the increasing rejection rate found with increasing timestep (accumulated time is equal to NT Delta tf where NT is the number of attempted moves, Delta t is the equivalent timestep and f is the fraction of attempted moves accepted). Nevertheless the lack of timestep drift in thermodynamic properties seen in SMC when compared with ED does offer significant advantages in the simulation of model colloidal liquids. The use of MC as a substitute for ED is not so advantageous as the rejection rate increases more dramatically with timestep than SMC. Also its formal relationship with BD/SMC for finite timesteps is not so clear.