Erratum and addenda: Solution of a new integral equation for pair correlation function in molecular liquids

1975 ◽  
Vol 62 (10) ◽  
pp. 4246-4246 ◽  
Author(s):  
L. J. Lowden ◽  
D. Chandler
Keyword(s):  
Integral Equation ◽  
Correlation Function ◽  
Pair Correlation Function ◽  
Pair Correlation ◽  
Molecular Liquids

scholarly journals The Kirkwood Superposition Approximation, Revisited and reexamined

2013 ◽  
Vol 1 (1) ◽  
pp. 27-35 ◽  
Author(s):  
Arieh Ben Naim
Keyword(s):  
Integral Equation ◽  
Correlation Function ◽  
Pair Correlation Function ◽  
Pair Correlation ◽  
Potential Of Mean Force ◽  
Superposition Approximation ◽  
Potential Of Mean Forces

The Kirkwood superposition approximation (KSA) was originally suggested to obtain a closure to an integral equation for the pair correlation function. It states that the potential of mean force of say, three particles may be approximated by sum of potential of mean forces of pairs of particles. Nowadays, this approximation is widely used, explicitly or implicitly, in many fields unrelated to the problem for which it was suggested.It is argued that the KSA is neither a good approximation nor a bad approximation; it is simply not an approximation at all.

Integral Equation Methods

1984 ◽  
Author(s):  
C. G. Gray ◽  
K. E. Gubbins
Keyword(s):  
Integral Equation ◽  
Perturbation Theory ◽  
Correlation Function ◽  
Pair Correlation Function ◽  
Infinite Order ◽  
Ad Hoc ◽  
Pair Correlation ◽  
Integral Equation Methods ◽  
Atomic Liquids ◽  
Integral Equation Approach

In this chapter we describe some of the integral equation methods which have been devised for calculating the angular pair correlation function g(rω1ω2) and the site-site pair correlation function gαβ( r ) for molecular liquids. These methods are in the main natural extensions of methods devised for calculating the pair correlation function g(r) for atomic liquids. They can be derived from infinite-order perturbation theory (an example is given in § 5.4.8), whereby one partially sums the perturbation series of Chapter 4 to infinite order usually with the help of diagrams, or graphs, but alternative methods of derivation are also available, e.g. functional expansions. The original integral equation theories are in a certain sense more complete than perturbation theories, in that the full correlation function g (or gαβ) is calculated, whereas in perturbation theory one calculates the correction g — g0 to the reference fluid value g0. On the other hand the perturbation theory approximations are controlled; one can estimate the error by calculating the next term. I t is extremely difficult to estimate a priori the error in integral equation approximations, since certain terms are neglected almost ad hoc. Their validity must therefore be a posteriori, according to agreement with computer simulation results (or, less satisfactorily, with experiment). Of particular interest are theories which are a combination of perturbation theory and an integral equation, which tend to have some of the advantages of both approaches (see also §5.3.1). An example is the GMF/SSC theory of §5.4.7. The structure of the integral equation approach for calculating g(r ω1 ω2) is as follows. One starts with the Ornstein-Zernike (OZ) integral equation (3.117) between the total correlation function h = g — 1 and the direct correlation function c, which we write here schematically as . . . h = c + pch (5.1) . . . or, even more schematically, as . . . h = h[c], (OZ) (5.2) . . . where h[c] denotes a functional of c.

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