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

$$C^{{\mbox{\tiny qm}}}_{{\mbox{\tiny E}}}(\tau) \; \equiv \;... ...\mathcal{F}(0) \mathcal{F}(\tau) \rangle_{{\mbox{\tiny E}}} ,$$ (2.46)

where the (generalized force) operator $\mathcal{F}$ is an abbreviation for $-\partial {\mathcal{H}}/\partial x$. Note that $\left\langle m\left\vert\mathcal{F}(0)\mathcal{F}(\tau)\right\vert m\right\rangle$ corresponding to the initial state (2.39) can be explicitly written
$\sum_n \vert{\mathcal{F}}_{nm}\vert^2 e^{-i\omega_{nm} \tau}$. Averaging over $M \gg 1$ adjacent initial states $\left\vert m\right\rangle$ (which nevertheless span a classically-small energy range) has been performed in order to extract the local average of the random values of $\vert{\mathcal{F}}_{nm}\vert^2$. This smearing will from now be implied by the microcanonical ensemble average at energy $E$, indicated by the notation $\langle \cdots \rangle_{{\mbox{\tiny E}}}$. The Fourier transform gives the auto-correlation spectrum

$$\tilde{C}^{{\mbox{\tiny qm}}}_{{\mbox{\tiny E}}}(\omega)= \... ...rt{\mathcal{F}}_{nm}\vert^2 2\pi\delta(\omega_{nm} - \omega) ,$$ (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 $\varepsilon > \Delta/\hbar$, where $\Delta$ is the mean level spacing in energy. Thus there are two components to the smoothing (`smearing') procedure: a smearing by width $M\Delta$ along the diagonal, and by width $\hbar \varepsilon$ off the diagonal. The smearing is chosen to be sufficient to allow $\tilde{C}^{{\mbox{\tiny qm}}}_{{\mbox{\tiny E}}}(\omega)$ to become a well-defined function of $\omega$, and smooth in $E$. By using the continuum limit substitution $\sum_n \rightarrow (1/\Delta) \int dE = (\hbar/\Delta) \int d\omega$, the auto-correlation spectrum can now be interpreted as giving a formula for the mean value of the squared matrix element:

$$\left\langle \left\vert {\mathcal{F}}_{nm} \right\vert^2 \r... ...(\omega = \omega_{nm}) . %%\hspace{.5in} \mbox{band profile}.$$ (2.48)

The right-hand side, viewed as a continuous function of $\omega$, is called the band profile^2.1of the matrix ${\mathcal{F}}_{nm}$ because it measures its off-diagonal (`band') structure. This construction is illustrated by Fig. 2.3. The matrix, and hence $\tilde{C}^{{\mbox{\tiny qm}}}_{{\mbox{\tiny E}}}(\omega)$, has structure on $\omega$-scales similar to inverse correlation times of the chaotic motion (for instance the bouncing rate in a billiard system). However the dependence on $E$ is very much weaker (if $\hbar/M\Delta$ 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,

$$\langle \Gamma_{nm}(\omega)\rangle \; = \; \frac{A^2}{4} \... ...[ \delta(\omega_{nm} - \omega) + \delta(\omega_{nm} + \omega)]$$ (2.49)

which is given by the band profile.