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

Abstract

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 a physical trajectory of a Brownian particle can be described by a random flight. Analysis of simple timelapses of physical trajectories (calculated over collisional timescales using a velocity autocorrelation function that captures the hydrodynamic and acoustic effects induced by the solvent) gave us two observations: (i) subsampled trajectories become genuinely memoryless only when the time step is ~200 times larger than the relaxation time, and (ii) the distribution of the step lengths has variances that remain significantly smaller than 2Dt (usually by a factor of ~2). This last fact is due to the combination of two effects: diffusional MSD is mathematically superballistic at short timescales and subsampled trajectories are moving averages of the underlying physical trajectory. In other words, the MSD of the physical trajectory at long time intervals asymptotically approaches 2Dt, but the MSD of the subsampled steps does not, even when the associated time interval is several hundred times larger than the relaxation time. I discuss how to best account for this effect in computational approaches.