Accurate Estimation\nof Diffusion Coefficients and\ntheir Uncertainties from Computer Simulation

Self-diffusion coefficients, <i>D</i>*, are\nroutinely\nestimated from molecular dynamics simulations by fitting a linear\nmodel to the observed mean squared displacements (MSDs) of mobile\nspecies. MSDs derived from simulations exhibit statistical noise that\ncauses uncertainty in the resulting estimate of <i>D</i>*. An optimal scheme for estimating <i>D</i>* minimizes\nthis uncertainty, i.e., it will have high statistical efficiency,\nand also gives an accurate estimate of the uncertainty itself. We\npresent a scheme for estimating <i>D</i>* from a single\nsimulation trajectory with a high statistical efficiency and accurately\nestimating the uncertainty in the predicted value. The statistical\ndistribution of MSDs observable from a given simulation is modeled\nas a multivariate normal distribution using an analytical covariance\nmatrix for an equivalent system of freely diffusing particles, which\nwe parametrize from the available simulation data. We use Bayesian\nregression to sample the distribution of linear models that are compatible\nwith this multivariate normal distribution to obtain a statistically\nefficient estimate of <i>D</i>* and an accurate estimate\nof the associated statistical uncertainty.