Reduced variance analysis of molecular dynamics simulations by linear combination of estimators

Building upon recent developments of force-based estimators with a reduced variance for the computation of densities, radial distribution functions, or local transport properties from molecular simulations, we show that the variance can be further reduced by considering optimal linear combinations of such estimators. This control variates approach, well known in statistics and already used in other branches of computational physics, has been comparatively much less exploited in molecular simulations. We illustrate this idea on the radial distribution function and the one-dimensional density of a bulk and confined Lennard-Jones fluid, where the optimal combination of estimators is determined for each distance or position, respectively. In addition to reducing the variance everywhere at virtually no additional cost, this approach cures an artifact of the initial force-based estimators, namely, small but non-zero values of the quantities in regions where they should vanish. Beyond the examples considered here, the present work highlights, more generally, the underexplored potential of control variates to estimate observables from molecular simulations.