There is a simple geometric derivation of the BBGKY hierarchy that is worthwhile discussing. Note that we cannot directly apply the notion of flow to the reduced density function. There can be multiple solutions to the dynamics at any given time which have the same first m coordinates. These solutions will evolve in separate directions, depending on the initial values of the n-m other coordinates. However, we may define a flow by averaging all of these solutions. The probability density flow at 
is given by 
, and the truncation of this to the m space of interest is 
. We average this over 
and define the reduced flow velocity through 
. With this flow defined, then we have from equation 5.3 the conservation equation
and making the substitution for the definition of 
puts the conservation equation in the form
Interchanging the 
and the integral is possible, and the resulting equation is easily seen to be the same as equation 5.9 (the quantity 
). Thus, the BBGKY equation for the reduced density function is simply the flow equation for the reduced density; the reduced dynamics is no longer Hamiltonian, but that of the superposition of Hamiltonian systems indicated by 
. This makes it quite clear why no equation involving only 
can succeed: the information from the variables 
affects the evolution at all times.