Variance in Free Energy Difference

The variance characterizes the spread of AA if an infinite number of independent simulations are carried out, each with a finite sample of size N. In practice, usually only one estimate (or a small number of repeats) of free energy differences are taken, and the variance in free energy must be estimated. One way to compute the variance is to use the error propagation formula (for a forward calculation) 

The scaled variance of ft A A, defined as Na AA, can be related to the probability distribution function of the perturbation x [26]  

This can be concluded from the Jarzynski equality (in the distribution function form) and the relationship between the / and g distributions. To repeat 

One can further conclude that that these two Gaussian distributions are symmetrically located on the upper and lower sides of AA, and the free energy difference A A, the mean work W OF, for the forward and — W 0 for the reverse transformation) and the variance of work obey the following relationships  

Now the variance in free energy difference is described in terms of the variance of work. The analysis above also indicates that the Gaussian distributions /(IF) and g W) are related 

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