Bias and error in estimates of equilibrium free-energy differences from nonequilibrium measurements

In 1997, Jarzynski proved a remarkable equality that allows one to compute the equilibrium free-energy difference DeltaF between two states from the probability distribution of the nonequilibrium work W done on the system to switch between the states, e(-DeltaF/kT) = , [Jarzynski, C. (1997) Phys. Rev. Lett. 87, 2690-2693]. The Jarzynski equality provides a powerful free-energy difference estimator from a set of N irreversible experiments and is closely related to free-energy perturbation, a common computational technique for estimating free-energy differences. Despite the many applications of the Jarzynski estimator, its behavior is only poorly understood. In this article we derive the large N limit for the Jarzynski estimator bias, variance, and mean square error that is correct for arbitrary perturbations. We then analyze the properties of the Jarzynski estimator for all N when the probability distribution of work values is Gaussian, as occurs, for example, in the near-equilibrium regime. This allows us to quantitatively compare it to two other free-energy difference estimators: the mean work estimator and the fluctuation-dissipation theorem estimator. We show that, for near-equilibrium switching, the Jarzynski estimator is always superior to the mean work estimator and is even superior to the fluctuation-dissipation estimator for small N. The Jarzynski-estimator bias is shown to be the dominant source of error in many cases. Our expression for the bias is used to develop a bias-corrected Jarzynski free-energy difference estimator in the near-equilibrium regime.