Accurate Estimation of Diffusion Coefficients and their Uncertainties from Computer Simulation

Self-diffusion coefficients, $D^$, are routinely estimated from molecular dynamics simulations by fitting a linear model to the observed mean-squared displacements (MSDs) of mobile species. MSDs derived from simulation exhibit statistical noise that causes uncertainty in the resulting estimate of $D^$. An optimal scheme for estimating $D^$ minimises this uncertainty, i.e., it will have high statistical efficiency, and also gives an accurate estimate of the uncertainty itself. We present a scheme for estimating $\D$ from a single simulation trajectory with high statistical efficiency and accurately estimating the uncertainty in the predicted value. The statistical distribution of MSDs observable from a given simulation is modelled as a multivariate normal distribution using an analytical covariance matrix for an equivalent system of freely diffusing particles, which we parameterise from the available simulation data. We use Bayesian regression to sample the distribution of linear models that are compatible with this multivariate normal distribution, to obtain a statistically efficient estimate of $D^$ and an accurate estimate of the associated statistical uncertainty.

💡 Research Summary

The paper addresses a fundamental problem in molecular dynamics (MD) simulations: the reliable estimation of the self‑diffusion coefficient D* from the mean‑squared displacement (MSD) data. The conventional practice of fitting a straight line to the MSD versus time and extracting the slope (via ordinary least squares, OLS) assumes that the data points are independent and identically distributed. In reality, MSD values are serially correlated and heteroscedastic, leading to statistically inefficient estimates and, more critically, to confidence intervals that are dramatically underestimated.

To overcome these limitations, the authors develop a Bayesian regression framework that attains near‑optimal statistical efficiency while providing an accurate quantification of uncertainty from a single simulation trajectory. The key steps are:

1. Modeling the MSD distribution – They treat the observed MSD vector x as a sample from a multivariate normal distribution with mean m (the linear model m(t)=6 D* t + c) and covariance Σ. Because the exact covariance is unknown, they construct an analytical model covariance Σ′ based on the theory of freely diffusing particles. The elements of Σ′ depend on the time‑dependent variances σ²