Consider Hamilton's equations in the form
We noted early in the course that an ensemble of systems evolving
according to these equations of motion would generate an
equilibrium ensemble (in this case, microcanonical). Recall that
the phase space distribution function 
satisfied
a Liouville equation:
where 
. We noted that if 
, then
f=f(H) is a pure function of the Hamiltonian which defined the
general class of distribution functions valid for equilibrium ensembles.
What does it mean, however, if 
? To answer this,
consider the problem of a simple harmonic oscillator. In an equilibrium ensemble
of simple harmonic oscillators at temperature T, the members of the ensemble will
undergo oscillatory motion about the potential minimum, with the amplitude of
this motion determined by the temperature. Now, however, consider driving
each oscillator with a time-dependent driving force F(t). Depending on how
complicated the forcing function F(t) is, the motion of each member of
the ensemble will, no longer, be simple oscillatory motion about the
potential minimum, but could be a very complex kind of motion that
explores large regions of the potential energy surface. In other words,
the ensemble of harmonic oscillators has been driven away from equilibrium
by the time-dependent force F(t). Because of this nonequilibrium behavior
of the ensemble, averages over the ensemble could become time-dependent quantities
rather than static quantities. Indeed, the distribution function , itself,
could be time-dependent. This can most easily be seen by considering the
equation of motion for a forced oscillator
The solution now depends on the entire history of the forcing function F(t), which can introduce explicit time-dependence into the ensemble distribution function.