Born-Green-Yvon hierarchy

This hierarchy is called the Born-Green-Yvon (BGY) hierarchy [6,7]. Again the specific case y = 2 is of most interest. 

The Born-Green-Yvon (BGY) hierarchy is truncated by introducinga super position approximation = correlation function 1234. The resulting pair of two simultaneous integral equations including the triplet- and pair-correlation functions 423 S i2> which hereafter... 

Finally, we mention that very recently three other integral equation approaches to treating polymer systems have been proposed. Chiew [104] has used the particle-particle perspective to develop theories of the intermolecular structure and thermodynamics of short chain fluids and mixtures. Lipson [105] has employed the Born-Green-Yvon (BGY) integral equation approach with the Kirkwood superposition approximation to treat compressible fluids and blends. Initial work with the BGY-based theory has considered lattice models and only thermodynamics, but in principle this approach can be applied to compute structural properties and treat continuum fluid models. Most recently, Gan and Eu employed a Kirkwood hierarchy approximation to construct a self-consistent integral equation theory of intramolecular and intermolecular correlations [106]. There are many differences between these integral equation approaches and PRISM theory which will be discussed in a future review [107]. 

In the following discussions, we use the expressions for gjj and g derived independently by Born and Green, Yvon, Kirkwood, and Bogoliubov in various different forms. These basically equivalent hierarchy of equations (sometimes known as either the BGY or the BBGKY hierarchy) can be expressed as an infinite set of the following integrodifferential equations N- 00) ... 

After Bogoliubov, Born, Green, Kirkwood, and Yvon, all of whom derived fEquation (172) is called the first hierarchy equation, Eq. (174) the second, etc. tThe virial expansion of the pressure is well-known example of such a series. 

It will be noted that the procedure used in obtaining Eq. (9) is identical with that followed in obtaining the hierarchy of equations of Bogoliubov, Born and Green, Kirkwood, and Yvon. There, however, the concern is with a distribution function / of low order, referring to one or a small number of molecules, while here the system of interest is of macroscopic size. Actually Eq. (9) could have been written down immediately, since it is known to be the form of the Liouville equation in the presence of non-conservative forces. (Cf. the discussion by Whitaker on integral invariants.)... 

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