The most general way a system can be driven away from equilibrium by
a forcing function 
is according to the equations of motion:
where the 3N functions 
and 
are required to satisfy the
incompressibility condition
in order to insure that the Liouville equation for 
is still valid.
These equations of motion will give rise to a distribution function
satisfying
with 
. (We assume that f is
normalized so that 
.)
What does the Liouville equation say about the nature of
in the limit that and are small, so that the displacement away
from equilibrium is, itself, small? To examine this question, we propose to
solve the Liouville equation perturbatively. Thus, let us assume
a solution of the form
Note, also, that the equations of motion 
take a perturbative
form
and as a result, the Liouville operator contains two pieces:
where 
and 
is assumed to satisfy
means the Hamiltonian part of the equations of motion
For an observable 
, the ensemble average of A is a time-dependent quantity:
which, when the assumed form for is substituted in, gives
where 
means average with respect to 
.