Lag time in diffusion coefficient estimation: Mean-squared-displacement versus variance methods in Brownian motion

Abstract

We investigate the relationship of the sampling time interval, Ï„ (where Ï„ >> Ï„_damp), to the possible difference of diffusion coefficient value, D, when measured from the variance of the normal-displacement-distribution (D_var) and from the slope of the linear mean-squared-displacement (MSD) against sampling time interval (D_msd). Consider the pure Brownian motion (BM) of solid 2-micrometer polystyrene spheres in water, where the viscosity is represented by the Langevin damping parameter of 0.262 µs at 298 Kelvin (K) temperature, observed for two total runtime durations: 1.0 x 10⁶ and 5 x 10⁶ time steps using Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS). This study calculates D_var and D_msd and compare them against the theoretical diffusion value of 0.245 µm²/s. For the measurement of both D_msd and D_var, the effect of the value of sampling time interval, Ï„, its simulation timestep, and dump interval on the precision and accuracy of diffusion value were observed. We conclude that the sampling time interval affects the precision and accuracy of the two methods, such that at a sampling time interval lower than the damping parameter, the two methods are more precise but are less accurate. The two methods, however, become more accurate at sampling time intervals higher than damping parameter, with the precision becoming lower at shorter trajectory lengths. This experiment concluded that accurate and precise diffusion coefficient measurements arise by ensuring that the sampling time interval is significantly greater than the damping parameter while maintaining long trajectory lengths.