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Generalized forceforce correlation and band profile
Here I show a way to express the average transition rates in terms of the
time correlation function

(2.46) 
where the (generalized force) operator is an abbreviation for
.
Note that
corresponding
to the initial state (2.39)
can be explicitly
written
.
Averaging over adjacent initial states
(which nevertheless span a classicallysmall energy range)
has been performed in order to extract the local average of
the random values of
.
This smearing will from now be implied by
the microcanonical ensemble average at energy , indicated
by the notation
.
The Fourier transform gives the autocorrelation spectrum

(2.47) 
which will play a central role in this
and the following chapters.
Here, as before, the delta function must be taken to have a width
, where is the mean level spacing in energy.
Thus there are two components to the smoothing (`smearing')
procedure: a smearing by width along the diagonal, and
by width
off the diagonal.
The smearing is chosen to be sufficient to allow
to become a welldefined function of ,
and smooth in .
By using the continuum limit substitution
, the autocorrelation spectrum
can now be interpreted as giving a formula for the mean value of the
squared matrix element:

(2.48) 
The righthand side, viewed as a continuous function of ,
is called the band profile^{2.1}of the matrix
because it measures its offdiagonal (`band') structure.
This construction is illustrated by Fig. 2.3.
The matrix, and hence
,
has structure on scales similar to inverse correlation times
of the chaotic motion
(for instance the bouncing rate in a billiard system).
However the dependence on is very much weaker
(if is chosen smaller than the shortest periodic orbit
[200], as discussed in Section 2.3.3),
only changing
over classicallylarge energy scales, so can be taken as constant in
any classicallysmall range.
An example matrix from a real system is shown in Fig. 2.7).
We can now replace the FGR transition
rate by its welldefined average over the width of the deltalike functions,

(2.49) 
which is given by the band profile.
Next: Stochastic energy spreading
Up: Review of the linear
Previous: Basis choice, perturbation theory
Alex Barnett
20011003