   Next: Stochastic energy spreading Up: Review of the linear Previous: Basis choice, perturbation theory

## 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 (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 (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: (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 , 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, (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