an ensemble of similar systems, appropriately chosen frequency averaging or local (spatial) averaging, for example. In addition, it is assumed that the initial density $\rho_0 = W_{\Gamma_0}$ is a smooth function on the scale of $\Delta x \Delta k = 1$.

For the full correlation function, we find that contributions to (\[G0\]) with $n\ne n'$ are removed by averaging so that we may assert in addition that $$\label{mean}
\langle W_\Gamma\rangle=\langle W_K\rangle \approx \rho.$$ The quantities $\hat{\Gamma}$ and $\hat{K}$ thus have the same mean when taking a suitable average and this mean value is given by the classical equilibrium (phase space) density $\rho$ obtained from (\[FPK\]), that is, $$\label{avGK}
\langle \hat{\Gamma}\rangle = \langle \hat{K} \rangle \approx W^{-1}[\rho].$$ We will find in the next section, however, that fluctuations of $\hat\Gamma$ about this mean are much stronger than those of $\hat{K}$, particularly in the limit of weak damping.

The averaged quantities in (\[avGK\]) contain detailed information about the system. Powerful numerical tools have been developed for computing $\rho$, and thus indirectly for the mean of $\hat{\Gamma}$. Among those, the DEA technique together with the DFM implementation on meshes is particularly suitable for complex structures, which has been used in a range of engineering 