In our discussion of special factorization schemes of the classical propagator for multiple time scale integration, we will consider a simple example of a single particle moving in one dimension with an equation of motion of the form
in which the force contains a rapidly and slowly varying contributions,
and 
, respectively. In order to construct an
integrator that exploits the separation of time scales, we introduce
a reference system given by the equations of motion:
The Liouville operator for this reference system is
Applying the Trotter theorem as in the last lecture, we can devise
an approximation propagation scheme for this reference system
over a time step, 
:
which will simply produce a velocity Verlet type evolution for
a time step .
The total Liouville operator for the system is
where
Note that the total propagator for a time step 
can be factorized
according to the breakup into a reference system and a correction 
:
which is accurate to order 
. Now, the middle propagator
can be handled by factorizing it as we did above and by introducing
a relationship between and , namely that 
,
where n is an integer. In this case, we may write
Substituting this into the factorization for the full propagator, we have
This constitutes a two time-step propgator, which uses a small
time step to integrate the fast force and a large
time step to integrate the slow force. Clearly, the
slow forces are updated n time less often than the
fast forces.
It is not necessary to derive a closed-form expression for the action of this propagator on the phase space vector, as we can use the direct translation technique of the last lecture. Translating each operator into an update step in pseudocode, we would write:
where dt = Dt/n. Such a propagator is called a ``reference system propagator algorithm'' or RESPA. It is easy to verify that it is both time-reversible and symplectic.