influence of higher order eigenvalues of the FP operator on the variances in Eqs. (\[TrK\]) and (\[TrG\]) such as through the classical decay of correlation contributions, Eq. (\[corr\]). The main result of this section is, however, the relation (\[TrG2\]) relating the fluctuations in the ’smooth’ part, $\hat{K}$, to the fluctuations in the total correlation function. Together with (\[avGK\]) and (\[tcorr\]), we can now relate [*the first and second moments of the distributions of both $\hat{\Gamma}$ and $\hat{K}$ to classical phase space observables*]{} such as the stationary phase space density $\rho$ and the classical phase space autocorrelation function $A_{\rm cl}(n)$. These quantities depend of course on the underlying classical dynamics.

Numerical illustration of fluctuations using quantum map models {#numsec}
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The fluctuations of $\hat{K}$ and $\hat{\Gamma}$ are now illustrated using a quantum map model. As asserted in the preceding discussion, averages (\[TrK\]) and (\[TrG\]) and the corresponding variances are insensitive to the underlying symmetries and to how chaotic or integrable the system is, although the detailed distributions are not.

![ \[fig:fluc\] Growth in the fluctuations of $\hat{K}$ and $\hat{\Gamma}$ are illustrated as $v\to 1$, measured respectively by $\Tr
     K^2$ and $\Tr \Gamma ^2$. Calculations are for a nearly Gaussian source density 