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

Analysis of the mean squared displacement of species k, ⟨r²_k⟩, 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 ⟨r²_k⟩(t) 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.