Modeling Brownian Motion as a Timelapse of the Physical, Persistent Trajectory

While it is very common to model diffusion as a random walk by assuming memorylessness of the trajectory and diffusive step lengths, these assumptions can lead to significant errors. This paper describes the extent to which the physical trajectory of a Brownian particle can be described by a random walk. Analysis of “timelapses” of physical trajectories (calculated over collisional time scales using a velocity autocorrelation function that captures the hydrodynamic and acoustic effects induced by the solvent) yielded two observations: (i) these subsampled trajectories become genuinely memoryless only when their time step is ∼200 times larger than the relaxation time, and (ii) the distributions of the subsampled step lengths have variances that are significantly smaller than the diffusional ones (usually by a factor of ∼2). This last observation is due to two facts: diffusional displacements are mathematically “superballistic” at short time scales, and subsampled trajectories are “moving averages” of the underlying physical trajectory. The counterintuitive result is that the mean squared displacement (MSD) of the physical trajectory asymptotically approaches 2Dt (where D is diffusivity) at long time intervals t, but the MSD of the individual subsampled steps does not, even when their duration is several hundred times larger than the relaxation time. I discuss how to best account for this effect in computational approaches.