BBGKY hierarchy

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) ... 

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... 

It can be shown with the help of the BBGKY hierarchy ofequationsthataknowledgeofthepO) andp functions, together with the pertinent principle of superimposition, is sufficient to produce functions and so on. 

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). 

Now, for so-called dilute gases we need only consider the behavior of any two molecules, i.e., we truncate the BBGKY hierarchy at s = 2 and set /s = 0. Thus, we also have... 

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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