outlined in the remainder of this section. In particular, if we average over a parameter such as frequency (or system dimension in the quantum map model), we can demonstrate that fluctuations in $\hat{\Gamma}$ scale simply with fluctuations in $\hat{K}$, according to $$\label{GvsK}
\langle \Tr \hat{\Gamma}^2 \rangle = \frac{1+v^2}{1-v^2}
\langle \Tr\hat{K}^2\rangle.$$ This expression is quite general and is equally valid in chaotic and integrable limits, for example. It is also true irrespective of whether the system has time reversal symmetry or not. It provides a direct quantitative measure of the greater fluctuations seen in $\hat{\Gamma}$, relative to those of $\hat{K}$, as $v\to 1$. Furthermore, we will see that the fluctuations of $\hat{K}$ itself can be obtained using information that is readily available from propagation of a corresponding classical density, as performed in complex structures using the DEA method, for example.

Using (\[matrixelement\]), the scaling relationship (\[GvsK\]) provides an equivalent scaling $$\label{varGKmatrix}
\langle |\Gamma_{ij}|^2 \rangle_N = \frac{1+v^2}{1-v^2}
\langle |K_{ij}|^2\rangle_N$$ of the relative sizes of corresponding matrix elements. It should be noted that in practice such averages over all matrix elements are dominated by diagonal elements in the weak damping limit or when the source intensity is extended over length scales much larger 