BBGKY hierarchy

In statistical physics, the BBGKY hierarchy (Bogoliubov–Born–Green–Kirkwood–Yvon hierarchy, sometimes called Bogoliubov hierarchy) is a set of equations describing the dynamics of a system of a large number of interacting particles. The equation for an "s"-particle distribution function (probability density function) in the BBGKY hierarchy includes the ("s" + 1)-particle distribution function, thus forming a coupled chain of equations. This formal theoretic result is named after Nikolay Bogolyubov, Max Born, Herbert S. Green, John Gamble Kirkwood, and .
Formulation.
The evolution of an "N"-particle system in absence of quantum fluctuations is given by the Liouville equation for the probability density function formula_1 in 6"N"-dimensional phase space (3 space and 3 momentum coordinates per particle)
where formula_3 are the coordinates and momentum for formula_4-th particle with mass formula_5, and the net force acting on the formula_4-th particle is
where formula_8 is the pair potential for interaction between particles, and formula_9 is the external-field potential. By integration over part of the variables, the Liouville equation can be transformed into a chain of equations where the first equation connects the evolution of one-particle probability density function with the two-particle probability density function, second equation connects the two-particle probability density function with the three-particle probability density function, and generally the "s"-th equation connects the "s"-particle probability density function 
with the ("s" + 1)-particle probability density function:
The equation above for "s"-particle distribution function is obtained by integration of the Liouville equation over the variables formula_12. The problem with the above equation is that it is not closed. To solve formula_13, one has to know formula_14, which in turn demands to solve formula_15 and all the way back to the full Liouville equation. However, one can solve formula_13, if formula_14 could be modeled. One such case is the Boltzmann equation for formula_18, where formula_19 is modeled based on the molecular chaos hypothesis (). In fact, in the Boltzmann equation formula_20 is the collision integral. This limiting process of obtaining Boltzmann equation from Liouville equation is known as Boltzmann–Grad limit.
Physical interpretation and applications.
Schematically, the Liouville equation gives us the time evolution for the whole formula_21-particle system in the form formula_22, which expresses an incompressible flow of the probability density in phase space. We then define the reduced distribution functions incrementally by integrating out another particle's degrees of freedom formula_23. An equation in the BBGKY hierarchy tells us that the time evolution for such a formula_13 is consequently given by a Liouville-like equation, but with a correction term that represents force-influence of the formula_25 suppressed particles 
The problem of solving the BBGKY hierarchy of equations is as hard as solving the original Liouville equation, but approximations for the BBGKY hierarchy (which allow truncation of the chain into a finite system of equations) can readily be made. The merit of these equations is that the higher distribution functions formula_27 affect the time evolution of formula_13 only implicitly via formula_29 Truncation of the BBGKY chain is a common starting point for many applications of kinetic theory that can be used for derivation of classical or quantum kinetic equations. In particular, truncation at the first equation or the first two equations can be used to derive classical and quantum Boltzmann equations and the first order corrections to the Boltzmann equations. Other approximations, such as the assumption that the density probability function depends only on the relative distance between the particles or the assumption of the hydrodynamic regime, can also render the BBGKY chain accessible to solution.
Bibliography.
"s"-particle distribution functions were introduced in classical statistical mechanics by J. Yvon in 1935. The BBGKY hierarchy of equations for "s"-particle distribution functions was written out and applied to the derivation of kinetic equations by Bogoliubov in the article received on July 1945 and published in 1946 in Russian and in English. The kinetic transport theory was considered by Kirkwood in the article received in October 1945 and published in March 1946, and in the subsequent articles. The first article by Born and Green considered a general kinetic theory of liquids and was received in February 1946 and published on 31 December 1946.