Probability Distribution Functions of Perturbations

The perturbation process can be quantified using distribution functions of the potential energy change AU (for FEP) or work W (for NEW). The procedure is the same for both the FEP and NEW calculations, thus we use the term perturbation and the notation x to unify both. 

An important feature for these distribution functions is that, instead of being independent of each other, they are coupled for a given pair of systems 0 and 1. For example, in an FEP calculation, we have [36, 37] 

In the right-hand side of the second equality, we have replaced Ui (r) with Un(r) + AU. It can be shown that for a NEW calculation, f(W) and g(W) obey the same relationship [7, 43], Thus, in unified notation, we have 

This equation relates the free energy difference between two systems to the individual perturbations x and the / and g distribution functions. The relationship (6.15) is important in both the characterization of free energy error and the development of improved free energy methods. 

An example plot of / and g distributions is shown in Fig. 6.4. Note that in general, the / and g distribution functions are not identical [cf. (6.15) one exception is the case of a NEW calculation with a reversible transition path, see Sect. 6.4.2]. They have different peak positions and widths (aj and a2g) characterizing the distributions. 

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Two concepts are helpful in understanding and characterizing relative free energy computational errors phase space relationships and probability distribution functions of perturbations. 

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