The observation from the Liouville equation that an isolated Hamiltonian system's entropy is constant (see section 5.1), and in fact that any volume integral of a point function of the density is constant, makes it simple to define the information flow into a subsystem. For if the subsystem were isolated, not coupled to the rest of the system, then the entropy of the subsystem would be a constant 
. When the subsystem is coupled, let the entropy be 
, a function of time. The negative rate of change of this entropy is the magnitude of information flow into the system, (negative flow indicates that the subsystem is losing information or ``structure'')
With 
we have immediately that
Note that the above expression is equivalent to 
since probability is conserved. If the evolution of the subsystem was completely specified by the Hamiltonian 
, substituting 
and using Gauss' theorem to integrate would give zero exactly. Since the flow is not completely determined by we substitute from the BBGKY equation 5.9 to find that the information change in the subsystem is due solely to the coupling of the subsystem to the rest of the system and is
With a slight adjustment to the derivation above we note that in fact the rate of change of any functional of a point function f of the density 
is given by the above with f' substituted for 
.