Overlapping phase space distributions

Obtaining a small difference in the results from the perturbations in both directions is a necessary but insufficient condition. It does not guarantee that phase space was adequately sampled. In practice, the efficiency of sampling is extremely difficult to determine. For similar reference and perturbed systems, considerable overlap of phase space distribution and a small free energy difference is expected. In practice, a limit of a few k T is usually imposed as a free energy difference that can be accurately calculated using the perturbation for-... 

Fig. 2.4. Schematic representation of the different relationships between the important regions in phase space for the reference (0) and the target (1) systems, and their possible interpretation in terms of probability distributions - it should be clarified that because AU can be distributed in a number of different ways, there is no obvious one-to-one relation between P0(AU), or Pi (AU), and the actual level of overlap of the ensembles [14]. (a) The two important regions do not overlap, (b) The important region of the target system is a subset of the important region of the reference system, (c) The important region of the reference system overlaps with only a part of the important region of the target state. Then enhanced sampling techniques of stratification or importance sampling that require the introduction of an intermediate ensemble should be employed (d)...

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. 

Eq. [13], These two functions have different ranges but are similar in form in regions of equal where the phase spaces of the two systems overlap. It can be shown that the excess chemical potential is related to these distribution functions by... 

As an example Fig. 6 shows the distribution of the ions for a potential difference of A(j) = 0(00) — 0(—00) = kT/cq between the two bulk phases. In these calculations the dielectric constant was taken as e = 80 for both phases, and the bulk concentrations of all ions were assumed to be equal. This simplifies the calculations, and the Debye length Lj), which is the same for both solutions, can be used to scale the v axis. The most important feature of these distributions is the overlap of the space-charge regions at the interface, which is clearly visible in the figure. 

In multicrystal diffraction, the diffraction of many small crystals in the three-dimensional space are compressed into one-dimensional, and lose the independence of all diffraction hkl, where the overlapping of the symmetry or the unsynunetry of diffraction peak obscures the profile of distribution curve of hkl diffraction intensity. So the abundant structural information contained in the powder diffraction pattern cannot be extracted, so for a long time the powder diffraction could only be used as the identification of phases. 

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