A time-independent free energy estimator for metadynamics

Metadynamics is a powerful and well-established enhanced sampling method for exploring and quantifying free energy surfaces of complex systems as a function of appropriately chosen variables. In the limit of long simulation time, metadynamics converges to the exact free energy surface plus a time-dependent constant. In this article, we analyze in detail this time-dependent constant. We show an easy way to calculate it, and by explicitly calculating the time dependence of this constant, we are able to derive a time-independent and locally convergent free energy estimator for metadynamics. We also derive an alternate procedure for obtaining the full unbiased distributions of generic operators from biased metadynamics simulations and explicitly test its usefulness.