Gore Jeff, Ritort Felix, Bustamante Carlos
Department of Physics, University of California, Berkeley, CA 94720, USA.
Proc Natl Acad Sci U S A. 2003 Oct 28;100(22):12564-9. doi: 10.1073/pnas.1635159100. Epub 2003 Oct 3.
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/kappaT =e-W/kappaT [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.
1997年，雅尔津斯基证明了一个引人注目的等式，该等式使人们能够根据系统在两个状态之间切换时所做的非平衡功(W)的概率分布来计算两个状态之间的平衡自由能差(\Delta F)，即(e^{-\Delta F/k_{B}T}=\langle e^{-W/k_{B}T}\rangle) [雅尔津斯基，C.（1997年）《物理评论快报》87，2690 - 2693]。雅尔津斯基等式从一组(N)个不可逆实验中提供了一个强大的自由能差估计器，并且与自由能微扰密切相关，自由能微扰是一种估计自由能差的常用计算技术。尽管雅尔津斯基估计器有许多应用，但其行为却鲜为人知。在本文中，我们推导了雅尔津斯基估计器偏差、方差和均方误差的大(N)极限，该极限对于任意微扰都是正确的。然后，当功值的概率分布为高斯分布时，例如在近平衡区域中出现的情况，我们分析了所有(N)时雅尔津斯基估计器的性质。这使我们能够将其与另外两个自由能差估计器进行定量比较：平均功估计器和涨落耗散定理估计器。我们表明，对于近平衡切换，雅尔津斯基估计器总是优于平均功估计器，并且在(N)较小时甚至优于涨落耗散估计器。在许多情况下，雅尔津斯基估计器的偏差被证明是误差的主要来源。我们的偏差表达式用于在近平衡区域中开发一种偏差校正的雅尔津斯基自由能差估计器。
