From the last lecture, we saw that Liouville's equation could be cast in the form
The Liouville equation is the foundation on which statistical mechanics rests. It will now be cast in a form that will be suggestive of a more general structure that has a definite quantum analog (to be revisited when we treat the quantum Liouville equation).
Define an operator
known as the Liouville operator ( 
- the i is there as a matter
of convention and has the effect of making L a Hermitian operator).
Then Liouville's equation can be written
The Liouville operator also be expressed as
where 
is known as the Poisson bracket between 
and 
:
Thus, the Liouville equation can be written as
The Liouville equation is a partial differential equation for the
phase space probability distribution function. Thus, it specifies a
general class of functions 
that satisfy it. In order to
obtain a specific solution requires more input information, such as
an initial condition on f, a boundary condition on f, and
other control variables that characterize the ensemble.