consistent with 
is substituted for 
above, i.e. substitute 
,
where 
,
for ,
then the subsystem density effectively decouples from the rest of the system.
The change in entropy of the subsystem is then zero. The result is more
general.
. (Factorization of complete density into subsystem and outside densities
implies no flow into subsystem). Let the complete distribution function
of n particles factor into
over the subsystem and 
over the rest of the system. Then the time derivative of any point function
of the subsystem reduced distribution function 
integrated over the subsystem variables has zero time derivative. i.e.
Proof: The time derivative may be written
Substitute for 
from the BBGKY equation 5.9,
The 
term of the resulting integral may be written as
where the last line, equation 5.18,
is found by integrating the line above it using Gauss' theorem. Note that
in applying Gauss' theorem, 
,
the velocity is not defined as 
because the Hamiltonian appearing is 
and not the full Hamiltonian 
.
Instead, the velocity is defined as 
.
The structure that makes 
is built into this definition because 
.
(This proves the statement leading to equation 5.14
that the flow into the subsystem from the
term is always zero.) The 
term may be rewritten as
where the last line, equation 5.19,
is found by integrating the inner integral using Gauss' theorem in a manner
similar to that leading to 5.18.
The sum of these terms is zero; therefore no information flows into the
subsystem when the distribution function factors into a subsystem part
and an outside part. QED.
Note that the theorem above does not imply that there will be no information flow into the subsystem for all time. It simply states that whenever the ensemble density factors that there is no flow at those times.
The theorem above leads to a somewhat surprising counter-intuitive result: coupling any subsystem density to a maximum entropy outside system density does not immediately lead to information flow to or from the subsystem. The corollary below immediately follows from the theorem above.
. (Maxent outside implies no information flow into subsystem). If
the complete distribution function factors as 
,
where ,
then there is no information flow into the subsystem.
Some comments on irreversibility and the equations of motion of the reduced density function are in order. These equations may form the basis for an approach to the understanding of the nature of irreversibility in thermodynamical systems. See, for example, [37]. Also [5] is an interesting discussion of the evolution of the correlation structure in Hamiltonian systems with two-body potentials. Other approaches exist, however. A proposal to understand the nature of irreversibility by modifying the equations of motion of physics is reviewed in [15]. Macroscopic coarse-graining plays a large role in some presentations of irreversibility, see for example [53].
It should be pointed out that studying irreversibility from the point of view of the ensemble density is somewhat flawed. In the case of the single universe that we live within there is only one system to consider. The key is to understand why irreversibility is a phenomenon in this universe.