In general, let 
be the density function describing the probabilities of seeing state x
at time t. Since the probability at any time of seeing some state
is one, we have 
.
Microscopically, consider a volume element V and the probability
in this element at time t, 
.
In general the time evolution of a set of many variables is not going to
be deterministic (in the sense that the future state of the system is not
specified by the present state of the system alone), such as in any relaxation
process driven by radiative cooling or by other interactions of the system
with a high complexity and uncertainty outside world. However, if the systems
described by the density function evolve deterministically and continuously
then it is possible to define a velocity field on the space of states 
.
The velocity field is simply given by the rate of change of system points.
Then, the change in the probability within V with respect to time
is due to the flow through the surface S of the volume element,
so we write
With the use of Gauss' theorem, the surface integral may be rewritten
and we find
Taking the small volume limit in equation 5.2,
the probability conservation equation, gives us the equation of continuity
So far this equation holds whenever a velocity field 
is definable. In the case of Hamiltonian evolution, it turns out that the
probability density behaves like an incompressible fluid. We show this
by considering Hamilton's equations 
and 
for each component of the coordinate and conjugate momentum. Operate with
the divergence so that
Then, noting that 
and making the substitutions from Hamilton's equations we find for Hamiltonian
evolution that 
and
where 
,
the sum over components being implicit. Finally, noting that we may take
the volume small, and noting that 
we find for Hamiltonian evolution
which is known as Liouville's equation. The total time derivative of
the density being zero indicates that the density at any time evolving
system point remains constant, and that in turn the volume integral of
any point function of the density, such as the entropy or information correlation
functions, remains constant.
In summary we have gone from a very general notion of probability conservation, through the addition of the information that the system evolves deterministically, to the additional constraint that the system evolves according to a Hamiltonian.