Multidimensional diffusion coefficient measurements using the ensemble-mean squared displacement method
We model pure Brownian motion (driftless and external-force free) of a probe particle by directly solving the Langevin equation in multiple dimensions: 1D (d = 1), 2D (d = 2), and 3D (d =3), where d is the number of degrees of freedom of the motion. We analyzed the values of the ensemble-averaged mean squared displacement (MSD) and the diffusion coefficient (D) from the different trajectories for different numbers of probes and lengths of the observation time (sampling period). We have observed that the 3D case provides the largest slope and smallest standard deviation of the time-dependent MSD behavior. Increasing the observation time and the number of probes improve the accuracy and precision of the D measurement. The 3D measurement analysis (d = 3) exhibited the lowest percent error with increasing number of steps and probes in comparison with the measurements at lower degrees of freedom. This study shows the advantage of using higher degrees of freedom (3D measurements) when analyzing Brownian motion using the ensemble-averaged MSD analysis technique.