Integrating the density function over a subset of the variables of dependence yields a reduced density function. In this section the subscript of the density function indicates its order. For example, for m<n, 
, where 
. The reduced density function also has a dynamics, which may be found by integrating the solution found from the Liouville equation.
Integrating both sides of equation 5.6 over 
yields what is known as the BBGKY hierarchy (Born, Bogoliubov, Green, Kirkwood, Yvon) [12]. First, take 
, where 
is the part of H that is the sum of functions of 
's and 
's with all indices 
, to find
Except for the last term, this equation is the Liouville equation for the reduced system with the reduced Hamiltonian. Because of this form, the last term of equation 5.9 is the only place where the reduced system couples to the rest of the system.
As an aside, in deriving equation 5.9 the following expression of the poisson bracket is useful.
The last summation always integrates to zero over .
In standard presentations (see, for example [69, 38, 57]), the BBGKY hierarchy is a set of equations which takes into account the information that the Hamiltonian has the form 
with 
and 
. If we use this information in equation 5.9 then we have 
for i<m, and the coupling of the reduced system to the rest of the system then explicitly occurs only through the potential function. Also for this case, when the distribution function is symmetric under permutation of the indices of its arguments, the coupling term reduces to an integral over 
with integrand involving 
.