Application to the Jarzynski Identity

In this section we explore the use of the skewed momenta method for estimating the equilibrium free energy from fast pulling trajectories via the Jarzynski identity [104]. The end result will be that generating trajectories with skewed momenta improves the accuracy of the calculated free energy. As described in Chap. 5, Jarzynski s identity states that 

With Jarzynski s identity in the form of (8.46), we can apply to it the skewed momenta method simply by setting A[r(f)] = exp(— /3Wt). However, we anticipate that the method will be most useful in the particular case of calculating free energy profiles from pulling experiments, for which Hummer and Szabo have provided a modified form of Jarzynski s expression [106]. 

The average defining the potential of mean force, (8.49), can be written as an average over a skewed distribution of initial momenta as described by (8.55). We can anticipate that skewed trajectories are associated with lower work, as the momenta can be biased so that important degrees of freedom tend to move in the same direction as the pulling potential. Specifically, the instantaneous contribution to the work of 

As an illustrative example of this approach, consider applying Jarzynski s identity to reconstruct a one-dimensional double-well energy profile of the form V(x), assumed unknown and which is to be recovered by the method, from pulling trajectories with a harmonic guiding potential 

A detailed numerical implementation of this method is discussed in [106]. W is the statistical weight of a trajectory, and the averages are taken over the ensemble of trajectories. In the unbiased case, W = exp -(3Wt), while in the biased case an additional factor must be included to account for the skewed momentum distribution W = exp(-/ Wt)w(p). Such simulations can be shown to increase accuracy in the reconstruction using the skewed momenta method because of the increase in the likelihood of generating low work values. For such reconstructions and other applications, e.g., to estimate free energy barriers and rate constants, we refer the reader to [117]. 

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