Two-body distribution function

The statistical mechanical value of <5Pw3> mdst be correctly weighted according to the properties of the radial distribution fimction g(r,2). The latter can be related to the two-body distribution function n (r, 2) through... 

There are two principal routes from the molecular distributions to the most important therm ynamic property of the interface—the surface tension. The first ( 4.4) is via the viriid t rem and requires a knowledge of the two-body distribution function g(ri2, Zj, Z2) the second... 

In Chapter 4 we obtained several equations which relate the density profile of a planar surface, p(z), to the pair potential, u(r), or to functionals of it such as the two-body distribution function or the direct correlation function c(r,2. Zi, z ). None of the equations, however, yields an explicit solution for p(z). in this chapter we describe some of the extra assumptions that have been made to enable them to be solved, and discuss the results for realistic forms of u(r). We consider primarily the Lennard-Jones (12,6) potential, (6.2), since it has been the most widely used, since there are several computer simulations for it, and since it is a reasonably realistic potential for simple fluids. ... 

If three-body collisions are neglected, which is permitted at sufficiently low densities, all the interactions take place between pairs of particles the two-particle distribution function will, therefore, satisfy Liouville s equation for two interacting particles. For /<2)(f + s) we may write Eq. (1-121) ... 

This binary collision approximation thus gives rise to a two-particle distribution function whose velocities change, due to the two-body force F12 in the time interval s, according to Newton s law, and whose positions change by the appropriate increments due to the particles velocities. 

A second, entirely different class of new polymer integral equation theories have been developed by Lipson and co-workers, Eu and Gan, " and Attard based on the site-site version of the Born-Green-Yvon (BGY) equation. The earliest work in this direction was apparently by Whittington and Dunfield, but they addressed only a special aspect of the isolated polymer problem (dilute solution). The central quantity in the BGY approaches is the formally exact expressions that relate two and three (or more) intramolecular and intermolecular distribution functions. The generalized site-site Ornstein-Zernike equations and direct correlation functions do not enter. In the BGY schemes the closure approximation(s) enter as approximate relations between the two- and three-body distribution functions supplemented with exact normalization and asymptotic conditions. In the recent BGY work of Taylor and Lipson a four-point distribution function also enters. 

The probability for a particular electron collision process to occur is expressed in tenns of the corresponding electron-impact cross section n which is a function of the energy of the colliding electron. All inelastic electron collision processes have a minimum energy (tlireshold) below which the process cannot occur for reasons of energy conservation. In plasmas, the electrons are not mono-energetic, but have an energy or velocity distribution,/(v). In those cases, it is often convenient to define a rate coefficient /cfor each two-body collision process ... 

After this computer experiment, a great number of papers followed. Some of them attempted to simulate with the ab-initio data the properties of the ion in solution at room temperature [76,77], others [78] attempted to determine, via Monte Carlo simulations, the free energy, enthalpy and entropy for the reaction (24). The discrepancy between experimental and simulated data was rationalized in terms of the inadequacy of a two-body potential to represent correctly the n-body system. In addition, the radial distribution function for the Li+(H20)6 cluster showed [78] only one maximum, pointing out that the six water molecules are in the first hydration shell of the ion. The Monte Carlo simulation [77] for the system Li+(H20)2oo predicted five water molecules in the first hydration shell. A subsequent MD simulation [79] of a system composed of one Li+ ion and 343 water molecules at T=298 K, with periodic boundary conditions, yielded... 

Expressions for the medium modifications of the cluster distribution functions can be derived in a quantum statistical approach to the few-body states, starting from a Hamiltonian describing the nucleon-nucleon interaction by the potential V"(12, l/2/) (1 denoting momentum, spin and isospin). We first discuss the two-particle correlations which have been considered extensively in the literature [5,7], Results for different quantities such as the spectral function, the deuteron binding energy and wave function as well as the two-nucleon scattering phase shifts in the isospin singlet and triplet channel have been evaluated for different temperatures and densities. The composition as well as the phase instability was calculated. 

For some time it has been known that the spectral moments, which are static properties of the absorption spectra, may be written as a virial expansion in powers of density, q", so that the nth virial coefficient describes the n-body contributions (n = 2, 3. ..) [400]. That dynamical properties like the spectral density, J co), may also be expanded in terms of powers of density has been tacitly assumed by a number of authors who have reported low-density absorption spectra as a sum of two components proportional to q2 and q3, respectively [100, 99, 140]. It has recently been shown by Moraldi (1990) that the spectral components proportional to q2 and q3 may indeed be related to the two- and three-body dynamical processes, provided a condition on time is satisfied [318, 297]. The proof resorts to an extension of the static pair and triplet distribution functions to describe the time evolution of the initial configurations these permit an expansion in terms of powers of density that is analogous to that of the static distribution functions [135],... 

Inspection of Fig. 1(c) reveals that there are a few pairs of atoms with a preferred distance. Analysis of many such images in terms of site occupation probabilities as a function of adatom distances revealed significant deviations from a random distance distribution, and the existence of adsorbate interactions which indeed oscillate with a wave vector of 2kp [16]. The decay followed the l/r2-prediction only for large distances, while significant deviations were observed at distances below 20 A and interpreted as a shortcoming of theory [16]. However, an independent study, carried out in parallel, focused on two body interactions only, i.e., the authors counted only those distances r from a selected atom to a nearby atom where no third scatterer (adatom or impurity) was closer than r [17]. This way, many body interactions were eliminated and the interaction energy E(r) yielded perfect... 

Figure 17. Radial distribution function for Kr at T — 297 K, calculated with the HMSA the dotted lines correspond to the two-body interaction and the full lines correspond to the two- plus three-body interaction. The variation g(xm), induced by the inclusion of the three-body interaction at the first maximum xm of g(r) is indicated for each density. Taken from Ref. [11].

Simulations of the liquid water properties have been the subject of many papers, see Ref. (374) for a review. Recently a two-body potential for the water dimer was computed by SAPT(DFT)375. Its accuracy was checked375 by comparison with the experimental second virial coefficients at various temperatures. As shown on Figure 1-16, the agreement between the theory and experiment is excellent. Given an accurate pair potential, and three-body terms computed by SAPT376, simulations of the radial 0-0, 0-H, and H-H distribution functions could be... 

With the above, a formal set of equations Is given, the elaboration of which requiring a solution for the problem that the recurrent relationships p p - p p, . .. diverge. Relatively simple densities, or distribution functions, are converted into more complex ones. A "closure" is needed to "stop this explosion". A number of such closures have been proposed, all involving an assumption of which the rigour has to be tested. Most of these write three-body interactions in terms of three two-body Interactions, weighted in some way. A well known example is Kirkwood s superposition closure, which reads ... 

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