We may also consider an ensemble of such systems for which there
are well defined macroscopic properties. This can be described by
means of a distribution function 
of phase
space points within the ensemble. By considering the number of
phase points entering and leaving an infinitesimal volume of
phase point [Tolman
[1962]], we get a generalized
form of Liouville's equation:
The operator 
is called the f-Liouvillean. Equation
(
) can be solved formally by integrating with
respect to t:
We can also consider the time dependence of a phase variable B as we follow a phase point through its trajectory:
The operator L is called the p-Liouvillean. This equation can be solved formally to give
The exponential of a Liouvillean is called a propagator.
From now on, the abbreviation 
will be used.
The p-Liouvillean and the f-Liouvillean are hermitian adjoints of each other:
If the system were described by a Hamiltonian, such as in
equation (), then Liouville's
theorem, 
, would hold, and equation () takes the
form of the usual Liouville equation. The existence of a
Hamiltonian is sufficient, but not necessary for this condition
to hold. In this case, the p- and f-Liouvilleans are identical
and self adjoint. Since we wish to describe systems in a
nonequilibrium steady state, the presence of dissipative terms
implies distinct p- and f-Liouvilleans, and the general equation
().
Macroscopic quantities are computed from microscopic quantities by means of phase averages, for example the temperature of the system ensemble (in equilibrium at least) is given by
We can determine the time evolution of the phase average by propagating the distribution function with the f-propagator, and then forming the phase average over a phase variable B:
By analogy with quantum mechanics, this is called the Schrödinger picture. Alternatively, one may form the phase average by following the phase variable along the trajectories:
This is the Heisenberg picture. The equivalence of the two
pictures is guaranteed by the adjointness property
().
