Estimation of Free-Energy Differences from Computed Work Distributions: An Application of Jarzynski’s Equality

Abstract

Equilibrium free-energy differences can be computed from nonequilibrium molecular dynamics (MD) simulations using Jarzynski's equality (Jarzynski, C. Phys. Rev. Lett.1997, 78, 2690) by combining a large set of independent trajectories (path ensemble). Here we present the multistep trajectory combination (MSTC) method to compute free-energy differences, which by combining trajectories significantly reduces the number of trajectories necessary to generate a representative path ensemble. This method generates well-sampled work distributions, even for large systems, by combining parts of a relatively small number of trajectories carried out in steps. To assess the efficiency of the MSTC method, we derived analytical expressions and used them to compute the bias and the variance of the free-energy estimates along with numerically calculated values. We show that the MSTC method significantly reduces both the bias and variance of the free-energy estimates compared to the estimates obtained using single-step trajectories. In addition, because in the MSTC method the process is divided into steps, it is feasible to compute the reverse transition. By combining the forward and reverse processes, the free-energy difference can be computed using the Crooks' fluctuation theorem (Crooks, G. E. J. Stat. Phys.1998, 90, 1481 and Crooks, G. E. Phys. Rev. E 2000, 61, 2361) or Bennett's acceptance ratio (Bennett, C. H. J. Comput. Phys. 1976, 22, 245), which further reduces the bias and variance of the estimates.