BBGKY hierarchy distribution functions

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. 

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... 

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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