Important Phase Space

So how is it possible to ensure this Examining the phase space overlap relationship of the reference and the target systems will give insight towards answering this question. 

A two-dimensional cartoon is helpful in understanding overlap relationships between the important phase spaces /, and /. As illustrated in Fig. 6.1, there are four possible ways that /, and / can be related (a) / can form a subset of /, (b) /, and r may almost coincide, i.e., have complete overlap (c) T( and / may have partial overlap and (d) / and r may have no overlap. For simplicity, in Fig. 6.1 we have only considered the case in which / is continuous in space but the same principle applies to more-complicated situations, in which regions of / are separated in space. 

Consider a single-stage FEP calculation with system 0 as the reference. For the situation illustrated in Fig. 6.1a, where T lies inside the T0, configurations in / are 

For the case in Fig. 6. la, the important configurations of an appropriately constructed intermediate, r, should be a subset of r and at the same time a superset of r, i.e., r0 g r M g r. Thus, two separate FEP calculations should be performed for the transformations 0 M and M — 1, yielding an overall free energy 

Finally, for the case Fig. 6.Id, in which Eg and r have no overlap, one may construct an intermediate whose important phase space regions contain both r0 and r, i.e., (Tq g r ) n (r G r j), and conduct two FEP simulations for M — 0 and M — 1. Correspondingly, the overall free energy difference is given 

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Fig. 6.1. A cartoon depiction of the phase space T and important phase space regions and their relationship. The important phase space regions of systems 0 (To ) and 1 (T) ) are abstractly represented by shaded and open oval shapes, respectively. These regions can be related in four ways (a) subset, (b) coincidence, (c) partial overlap, and (d) no overlap. Also sketched is the important phase space region of the intermediate M in a two-stage calculation (see section on Multiple-Stage Design ). The appropriate staging strategy differs according to the different overlap relationships between F0 and 77...

As the initial and final states are set by the problem under study, their important phase space relationship could be any one of the cases illustrated in Fig. 6.1. For cases Fig. 6.1c, d, it is impossible to construct a funnel path from 0 to 1 directly. To satisfy the funnel requirement, similar to the MFEP calculation, a staged NEW calculation can be performed. For example, in the case Fig. 6.1c, one can first construct an intermediate in the common region of / ,[ and /), then perform two separate NEW calculations following the paths 0 —> M and 1 —> M, respectively. This NEW-overlap sampling (NEW-OS) technique will be discussed in detail in Sect. 6.6. 

Fig. 6.3. To ensure the accuracy of a nonequilibrium work free energy calculation, the switching paths should go down the funnel. The important phase space regions for the intermediate states along the ideal funnel paths are illustrated in this plot, for the case where r0 and / are partially overlapped. Two funnel paths need to be constructed to transfer the systems from both 0 and 1 to a common intermediate M where rm is inside the r0 and J overlap region. The construction of such paths is discussed in Sect. 6.6...

It would be valuable if one could proceed with a reliable free energy calculation without having to be too concerned about the important phase space and entropy of the systems of interest, and to analyze the perturbation distribution functions. The OS technique [35, 43, 44, 54] has been developed for this purpose. Since this is developed from Bennett s acceptance ratio method, this will also be reviewed in this section. That is, we focus on the situation in which the two systems of interest (or intermediates in between) have partial overlap in their important phase space regions. The partial overlap relationship should represent the situation found in a wide range of real problems. 

As concluded in Sect. 6.3.1, when the important phase spaces // and r have partial overlap, the appropriate staging strategy is to construct an intermediate M whose... 

Now we proceed with the construction of the intermediate M and the switching paths 0 —> M and 1 —> M. The key considerations are (1) both paths should follow the funnel sampling path to eliminate the systematic error due to the inaccessibility of important phase space, and (2) M is the optimal choice for minimizing the variance of the calculation. The optimal intermediate (for minimal variance) can be defined as follows [43] ... 

In research similar to that described in section 4.3.1, phase space structures and phase space bottlenecks have been used to analyze unimolecular reaction dynamics (Davis and Gray, 1986 Gray et al., 1986b Gray and Rice, 1987 Zhao and Rice, 1992 Jain et al., 1993 DeLeon, 1992a,b Davis and Skodje, 1992). Important phase space structural properties are illustrated in figure 8.11, for the one-dimensional pendulum Hamiltonian (Lichtenberg and Lieberman, 1991) ... 

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