BBGKY hierarchy

In statistical physics the BBGKY hierarchy describes the dynamics of systems consisting of a large number of interacting particles. Subsequent equations in the BBGKY hierarchy express the evolution of the N-particle distribution function in terms of the (N+1)-particle distribution function. This formal theoretic result is named after Bogoliubov, Born, Green, Kirkwood, and Yvon.

The kinetics of an N-particle system is given by the Liouville equation operating on a probability density function in 6N phase space. The BBGKY hierarchy can be used to cast the Liouville equation into a chain of equations where the first equation connects the evolution of one-particle density probability with the two-particle density probability function, and similarly the i-th equation connects the i-th particle and (i+1)-st particle density probability function. The problem of solving the BBGKY chain is as hard as solving the original Liouville equation, but approximations for the BBGKY chain can readily be made. Truncation of the BBGKY chain (usually at the level of the first equation or the first two equations) is a common starting point for many applications of kinetic theory.Some other approximations, such as the assumption that the density probability is function of only the relative distance, and working in hydrodynamic regime, can render all levels of equations of the BBGKY chain accessible to solution.