Pair action

The Hopewell peoples did produce finely twined fabrics of various types (12,18). Church (18) indicates that the most common type found in three Ohio Hopewell mounds is a spaced alternate-pair weft twining. One instance of alternate-pair action was observed on the Tunacunnhee Mound C earspool, but the general corrosion of the earspool precluded clear identification. 

In practice we move all slices of all protons at the same time by a simple random move in a box and we pre-reject the moves with the effective pair action. 

The other main avenue to tackle the A-particle quantum problem with Pis focuses on the so-called pair product actions. These are approaches that have been proven to be very cost effective, as relatively low P discretizations work excellently well. Besides, the use of pair actions has led to a deep understanding of helium and the system of hard spheres at low temperatures [83-108]. Clear indications of how to proceed along this line were given by Barker in his pioneering work on quantum hard spheres (QHS) [24], Klemm and Storer [83], and Ceperley [28]. Essentially, this sort of approach consists of representing the V-particle density matrix through a product of density matrices, which includes those of the free particles and those of the reduced masses of every pair of particles. The potential is assumed pair-wise additive and, for definiteness, P time slices in... 

For systems ruled by continuous interaction potentials (e.g., helium) a fine justification of Eq. (44) can be found in Ref. 28, where the Feynman-Kac formula is used. Accordingly, the pair action in these cases can be written as... 

Very elaborate methods to fix are available [28]. On the other hand, for potentials with an infinite discontinuity (e.g., hard cores) special propagators are needed, but the general pair action scheme contained in Eqs. (42)-(45) still holds and can be employed advantageously. The crucial point is that, although there is a substantial increase in the number of calculations to be performed in a simulation using with respect to those of the primitive propagator, the reduction that... 

A full discussion of the reasons for the success of the pair action approach, some estimates of the order of accuracy, and the measures to enhance its performance can be found in Ceperley s review [28], However, the exact density matrix contains not only pair terms but also triplet terms, quadruplet terms, and so on. These terms are needed in the study of situations at very high densities. In relation to this, although some clever tricks can be used to deal with the pair case, the next extension to deal with triplet terms is far from straightforward, because of the increase in the number of variables needed (e.g., for a homogeneous and isotropic fluid from four variables in the pair case to 13 in the triplet case) [28]. Furthermore, it is difficult to anticipate what the order of accuracy ) of a triplet action will be. This situation contrasts sharply with that of the higher-order propagators, for which the order of accuracy is always known. Another problem in this context is that the usefulness of pair actions cannot be extended to deal with non-symmetri-cally spherical potentials (e.g., molecular fluids). 

To get a more definite idea of the form of a partition function when pair actions are made, it may be helpful to consider the one arising from the JOHS and CBHS propagators. It is the same as that written in Eq. (25b), but including as the effective potential W p the expression... 

When utihzing PI simulations, the actual function can be approximated by increasing the discretization of the necklaces, as discus before. Therefore, for the primitive, pair action, and SCVJ propagators, it is straightforward to estabhsh... 

In Eq. (78) stands for any of the corresponding PI effective potentials utilized to deal with the set of NxP beads. Also, note that the beads considered in the tj-sum are those significant to compute structures, that is, with (X = P,Atg =tg) the P equivalent beads for the primitive and the pair action propagators arise... 

By continuing within the framework of PI calculations with the primitive, pair action, and SCVJ propagators, the next step is to consider the action of a weak continuous field T and obtain the responses from the fluid to variations <5 P [25, 35, 36]. T is assumed to be compatible with the existence of eigenstates in the fluid and is defined to act on each particle as P(rp. The Hamiltonian for N particles is then... 

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