The Green-Kubo (GK) formalism is a cornerstone of non-equilibrium statistical mechanics. It relates transport phenomena to spontaneous time-correlated fluctuations of microscopic properties, which can be simulated using molecular dynamics (MD). The GK formalism has enabled the prediction and understanding of physicochemical properties used in diverse scientific and engineering disciplines, including diffusivity, ionic conductivity, and viscosity.
Historically, algorithms for computing transport properties have been based either on numerical quadrature of an autocorrelation function (Green–Kubo methods) or regression of a straight line to a mean square displacement curve (Einstein–Helfand methods). While these two methods are formally equivalent, the algorithms differ in practice. Both methods share the limitation that they process time-dependent quantities (autocorrelation functions or mean square displacements) that have correlated uncertainties at different time lags. These correlated errors are difficult to quantify and are rarely considered, resulting in unreliable uncertainty estimates of the computed transport properties.
We developed a new approach for computing transport properties from MD simulations, based on the spectral analysis of the relevant time series. The resulting algorithm, called the Stable AutoCorrelation Integral Estimator (STACIE), provides statistically consistent estimates of transport properties, including reliable uncertainty estimates. It has been validated on a broad range of transport properties. Furthermore, we tested STACIE on a newly developed massive validation set, called the AutoCorrelation Integral Drill (ACID), which comprises 15360 test cases for which the expected autocorrelation integral is known exactly (and conveniently equal to one). The results of this validation confirm the effectiveness of STACIE. Traditional methods cannot even be applied to such large volumes of data because they require manual choices, such as identifying a region with a slope of one in the log-log plot of the mean square displacements. Besides transport properties, STACIE can also estimate the exponential correlation time, which corresponds to the slowest relaxation mode of a molecular model. In ongoing research, we are extending STACIE to compute cross-transport properties (such as thermoelectric coefficients), handle correlations between different input time series and apply similar algorithms to non-equilibrium MD simulations. In addition, a robust equilibration detection algorithm is being developed, building on STACIE's robust uncertainty quantification. We are using STACIE to study the transport properties of lubricants, slags, hydrogels, hydrated silicate ionic liquids and solid electrolytes.
Core publications
STable AutoCorrelation Integral Estimator (STACIE): Robust and accurate transport properties from molecular dynamics simulations. G. Toraman, D. Fauconnier, and T. Verstraelen (2025) Computational Chemistry, 65 (19): 10445-10464. doi: 10.1021/acs.jcim.5c01475
