Recall that the Fourier transform of a time correlation function can be related to some kind of frequency spectrum. For example, the Fourier transform of the velocity autocorrelation function of a particular degree of freedom q of interest
where 
,
gives the relevant frequencies contributing to the dynamics of q, but
does not give amplitudes. This ``frequency'' spectrum 
is simply
given by
That is, we take the Laplace transform of 
using 
.
Since carries information about the relevant frequencies of the
system, the decay of in time is a measure of how strongly coupled
the motion of q is to the rest of the bath, i.e., how much of an overlap
there is between the relevant frequencies of the bath and those of q.
The more of an overlap there is, the more mixing there will be between the system
and the bath, and hence, the more rapidly the motion of the system will become
vibrationally ``out of phase'' or decorrelated with itself. Thus, the decay
time of , which is denoted 
is called the vibrational dephasing
time.
Another measure of the strength of the coupling between the system and the bath is the time required for the system to dissipate energy into the bath when it is excited away from equilibrium. This time can be obtained by studying the decay of the energy autocorrelation function:
where 
is defined to be
The decay time of this correlation function is denoted 
.
The question then becomes: what are these characteristic decay times and how are they related? To answer this, we will take a phenomenological approach. We will assume the validity of the GLE for q:
and use it to calculate
and .
Suppose the potential 
is harmonic and takes the form
Substituting into the GLE and dividing through by m gives
where
An equation of motion for can be obtained directly by
multiplying both sides of the GLE by 
and averaging over
a canonical ensemble:
Recall that
and note that
also
Thus,
Combining these results gives an equation for
which is known as the memory function equation and the kernel K(t) is known as the memory function or memory kernel. This type of integro-differential equation is called a Volterra equation and it can be solved by Laplace transforms.
Taking the Laplace transform of both sides gives
However, it is clear that 
and also
Thus, it follows that
In order to perform the inverse Laplace transform, we need the poles of the integrand, which will be determined by the solutions of
which we could solve directly if we knew the explicit form of 
.
However, if 
is sufficiently larger than 
, then it is
possible to develop a perturbation solution to this equation. Let us assume
the solutions for s can be written as
Substituting in this ansatz gives
Since we are assuming 
is small, then to lowest order, we have
so that 
. The first order equation then becomes
or
Note, however, that
Thus, stopping the first order result, the poles of the integrand occur at
Define
Then
and is then given by the contour integral
Taking the residue at each pole, we find
which can be simplified to give
Thus, we see that the GLE predicts oscillates with a
frequency 
and decays exponentially. From the exponential decay, we can
directly read off the time :
That is, the value of the real part of the Fourier (Laplace) transform of
the friction kernel evaluated at the renormalized frequency divided by 2m
gives the vibrational dephasing time! By a similar scheme, one can easily
show that the position autocorrelation function 
decays with the same dephasing time. It's explicit form is
The energy autocorrelation function 
can be expressed in terms of
the more primitive correlation functions 
and . It is
a straightforward, although extremely tedious, matter to show that the
relation, valid for the harmonic potential of mean force, is
Substituting in the expressions for and gives
so that the decay time can be seen to be
and therefore, the relation between and can be seen immediately to be
The incredible fact is that this result is also true quantum mechanically. That is, by doing a simple, purely classical treatment of the problem, we obtained a result that turns out to be the correct quantum mechanical result!
Just how big are these times? If is very large compared to
any typical frequency relevant to the bath, then the friction kernel evaluated
at this frequency will be extremely small, giving rise to a long decay time.
This result is expect, since, if is large compared to the bath, there are
very few ways in which the system can dissipate energy into the bath.
The situation changes dramatically, however, if a small amount of
anharmonicity is added to the potential of mean force.
The figure below illustrates the point for a harmonic diatomic molecule
interacting with a Lennard-Jones bath. The top figure shows the
velocity autocorrelation function for an oscillator whose frequency is
approximately 3 times the characteristic frequency of the bath, while the
bottom one shows the velocity autocorrelation function for the
case that the frequency disparity is a factor of 6.
