This paper is concerned with fitting the mean-square displacement (MSD) function, and extract reliable and accurate values for the diffusion coefficient D. In this work, we present a new optimal and robust nonlinear regression model capable of fitting the MSD function with different regimes corresponding to different time scales. The algorithm presented here achieves two major goals; a more accurate estimation of D as well as extracting information about the short time behavior. The algorithm fits the MSD to a continuous piece-wise function and predicts all the coefficients in the model including the breakpoints. The novelty of this approach lies in its ability to find the breakpoints which separate different modes of motion. We tested our algorithm using numerical experiments, and our fits described the data remarkably well. In addition, we applied our algorithm to extract D for water based on Molecular Dynamics (MD) simulations. The results of our fits are in good agreement with the experimentally reported values.
Riahi, M. K., Qattan, I. A., Hassan, J., & Homouz, D. (2019). Identifying short- and long-time modes of the mean-square displacement: An improved nonlinear fitting approach. AIP Advances, 9(5). https://doi.org/10.1063/1.5098051