Bias and error in estimates of equilibrium free-energy differences from nonequilibrium measurements
- Jeff Gore*,
- Felix Ritort*,†, and
- Carlos Bustamante*,‡,§,¶
- *Department of Physics, University of California, Berkeley, CA 94720; †Department of Physics, University of Barcelona, Diagonal 647, 08028 Barcelona, Spain; ‡Department of Molecular and Cell Biology and Howard Hughes Medical Institute, University of California, Berkeley, CA 94720; and §Physical Biosciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720
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Contributed by Carlos Bustamante, August 12, 2003
Abstract
In 1997, Jarzynski proved a remarkable equality that allows one to compute the equilibrium free-energy difference ΔF between two states from the probability distribution of the nonequilibrium work W done on the system to switch between the states, e –Δ F /kT = 〈e – W /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.
