BBGKY equation

A structure of the obtained set of equations derived by us in [81, 86] is very close to the famous BBGKI set of equations widely used in the statistical physics of dense gases and liquids [76]. Therefore, we presented the master equation of the Markov process in a form of the infinite set of deterministic coupled equations for averages (equation (2.3.34)). Practical use of these equations requires us to reduce them, retaining the joint correlation functions only. 

The analogy just mentioned with the BBGKI set of equations being quite prominent still needs more detailed specification. To cut off an infinite hierarchy of coupled equations for many-particle densities, methods developed in the statistical theory of dense gases and liquids could be good candidates to be applied. However, one has to take into account that a number of the... 

Equation (1.24) is very similar to that of the single-particle distribution function of classical statistical mechanics. In the limit h—>0 we get the first equation of the BBGKY hierarchy. 

Having substituted Eq. (27) into the second equation of the BBGKY hierarchy at equilibrium [Eq. (23)], we arrived at... 

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) ... 

A realistic theory of nematics should, of course, incorporate the attractive potential between the molecules as well as their hard rod features. There have been several attempts to develop such hybrid models. Equations of state have been derived based on the Percus-Yevick and BBGKY approximations for spherical molecules subject to an attractive Maier-Saupe potential.However, a drawback with these models is that they lead to y = 1 (see (2.3.18)). 

Statistical mechanics when based on Liouville s theorem yields a hierarchy of equations (BBGKY hierarchy) that makes use of the 5-particle distribution function /<)< giving the probability of finding s particles, i = 1... j, out of the N particles in the system in the positions ri r, and... 

Equation (3.24) is the reduced Liouville equation for pairwise additive interaction forces. Note that this is an integro-differential equation, where the evolution of the fs distribution depends on the next higher-order fs+i distribution. This is known as the BBGKY hierarchy (named after its originators Bogoliubov, Bom, Green, Kirkwood, and Yvon see the Further Reading section at the end of this chapter). 

Equations (4.52) and (4.53) illustrate the BBGKY hierarchy for the equilibrium configurational distribution functions some truncation (closure) is necessary to obtain solutions. For example, neglecting three-body effects by setting cb(ri, rs) = 0, gives from Eq. (4.52) the dilute gas... 

One can proceed by deriving the equation of motion for the two-particle distribution function and so on. Continuing this way, one finds the well-known BBGKY hierarchy, in which distribution functions of higher order enter successively. A detailed study of the two-particle Wigner distribution... 

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