More efficient Brownian dynamics algorithms

We examine the relative efficiencies of three algorithms for performing Bro wnian Dynamics simulations without many-body hydrodynamics. We compare the conventional Brownian Dynamics algorithm of Ermak (CBD), Smart Monte Carlo (SMC) which incorporates Boltzmann sampling into essentially a CBD procedur e, and the Stochastic Runge Kutta (SRK) method. We show, using the repulsiv e potential phi (r) = epsilon(sigma /r)", where n = 36 and 72, that the SRK algorithm gives the most accurate short-time dynamics for the mean-square displacements. The SRK algorithm static and dynamical properties converge b etter with a reducing time step to the exact values, than those generated b y the CBD algorithm; giving efficiency gains typically of a factor of 3-4. Both CBD and SMC have the incorrect sign for the first correction term to t he mean square displacement in a time step, whereas the SRK algorithm gives essentially the exact solution to order Deltat(2), where Deltat is the sim ulation time step. In fact, these correction terms are almost equal and opp osite in sign. Expressions for these terms were derived in terms of the ave rage interaction energy per particle. The force, shear and bulk stress auto correlation f unctions were calculated. The average energy per particle and time correlation f unctions at short time have values in excess of the exa ct values, while the corresponding quantities for SRK are below this. This difference in behaviour can be traced back to the extent of compliance of t he particle trajectories with the exact expansion of the Smoluchowski equat ion. The accuracy, at a given value of the time step, of the stochastic alg orithms can significantly depend on the form of the interaction potential b etween particles. It is also demonstrated that the long time limits of vari ous correlation functions are fairly insensitive to a particular scheme (SR K or CBD) used in the simulations. All the correlation functions have a str etched exponential region at intermediate to long times, and the values of the exponents on density and force law steepness have been determined.