The convolution integral term
is called the memory integral because it depends, in general, on the
entire history of the evolution of q. Physically it expresses the fact
that the bath requires a finite time to respond to
any fluctuation in the motion of the system (q). This, in turn, affects how the
bath acts back on the system. Thus, the force that the bath exerts on the
system presently depends on what the system coordinate q did in the past.
However, we have seen previously the regression of fluctuations (their decay
to 0) over time. Thus, we expect that what the system did very far in the past
will no longer the force it feels presently, i.e., that the lower limit of
the memory integral (which is rigorously 0) could be replaced by
, where 
is the maximum time over which
memory of what the system coordinate did in the past is important. This can
be interpreted as a indicating a certain decay time for the friction kernel 
.
In fact, often does decay to 0 in a relatively short time. Often this decay
takes the form of a rapid initial decay followed by a slow final decay, as shown in
the figure below:
Consider the extreme case that the bath is capable of responding infinitely
quickly to changes in the system coordinate q. This would be the case, for
example, if there were a large mass disparity between the system and the bath
( 
). Then, the bath retains no memory of what the system did
in the past, and we could take to be a 
-function in time:
Then
and the GLE becomes
This simpler equation of motion is known as the Langevin equation and it is
clearly a special case of the more generalized equation of motion. It is often
invoked to describe brownian motion where clearly such a mass disparity is
present. The constant 
is known as the static friction and is given by
In fact, this is a general relation for determining the static friction constant.
The other extreme is a very sluggish bath that responds slowly to changes in
the system coordinate. In this case, we may take to be a constant
, at least, for times short compared to the
response time of the bath. Then, the memory integral becomes
and the GLE becomes
where the friction term now manifests itself as an extra harmonic term added to the potential. Such a term has the effect of trapping the system in certain regions of configuration space, an effect known as dynamic caging. An example of this is a dilute mixture of small, light particles in a bath of heavy, large particles. The light particles can get trapped in regions of space where many bath particles are in a sort of spatial ``cage.'' Only the rare fluctuations in the bath that open up larger holes in configuration space allow the light particles to escape the cage, occasionally, after which, they often get trapped again in a new cage for a similar time interval.