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Generalized force-force correlation and band profile
Here I show a way to express the average transition rates in terms of the
time correlation function
![\begin{displaymath}
C^{{\mbox{\tiny qm}}}_{{\mbox{\tiny E}}}(\tau) \; \equiv \;...
...\mathcal{F}(0) \mathcal{F}(\tau)
\rangle_{{\mbox{\tiny E}}} ,
\end{displaymath}](img297.gif) |
(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 classically-small 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 auto-correlation spectrum
![\begin{displaymath}
\tilde{C}^{{\mbox{\tiny qm}}}_{{\mbox{\tiny E}}}(\omega)= \...
...rt{\mathcal{F}}_{nm}\vert^2 2\pi\delta(\omega_{nm} - \omega) ,
\end{displaymath}](img303.gif) |
(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 well-defined function of
,
and smooth in
.
By using the continuum limit substitution
, the auto-correlation spectrum
can now be interpreted as giving a formula for the mean value of the
squared matrix element:
![\begin{displaymath}
\left\langle \left\vert {\mathcal{F}}_{nm} \right\vert^2 \r...
...(\omega = \omega_{nm}) .
%%\hspace{.5in} \mbox{band profile}.
\end{displaymath}](img307.gif) |
(2.48) |
The right-hand side, viewed as a continuous function of
,
is called the band profile2.1of the matrix
because it measures its off-diagonal (`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 classically-large energy scales, so can be taken as constant in
any classically-small range.
An example matrix from a real system is shown in Fig. 2.7).
We can now replace the FGR transition
rate by its well-defined average over the width of the delta-like functions,
![\begin{displaymath}
\langle \Gamma_{nm}(\omega)\rangle \; = \;
\frac{A^2}{4} \...
...[ \delta(\omega_{nm} - \omega) + \delta(\omega_{nm} + \omega)]
\end{displaymath}](img310.gif) |
(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
2001-10-03