A general expression for the statistical error in a diffusion coefficient obtained from a solid-state molecular-dynamics simulation

Abstract

Analysis of the mean squared displacement of species $k$ , $\left(r_k^2\right)$ , as a function of simulation time $t$ constitutes a powerful method for extracting, from a molecular-dynamics (MD) simulation, the tracer diffusion coefficient, $D_k^{*}$ . The statistical error in $D_k^{*}$ is seldom considered, and when it is done, the error is generally underestimated. In this study, we examined the statistics of $\left(r_k^2\right)\left(t\right)$ curves generated by solid-state diffusion by means of kinetic Monte Carlo sampling. Our results indicate that the statistical error in $D_k^{*}$ depends, in a strongly interrelated way, on the simulation time, the cell size, and the number of relevant point defects in the simulation cell. Reducing our results to one key quantity-the number of $k$ particles that have jumped at least once-we derive a closed-form expression for the relative uncertainty in $D_k^{*}$ . We confirm the accuracy of our expression through comparisons with self-generated MD diffusion data. With the expression, we formulate a set of simple rules that encourage the efficient use of computational resources for MD simulations.

Keywords: diffusion; kinetic Monte Carlo, kMC; mean squared displacement, MSD; molecular dynamics, MD; statistical error.