= \frac{1}{1- v^4} \sum_{n=-\infty}^\infty v^{2|n|} {A}(n)\, .$$ So far the calculation is exact and completely independent of the classical limit.

The quantity ${A}(n)$ can again be related to the classical phase space dynamics using $$A(n) = {\rm Tr}\left( \hat{\Gamma}_0 \hat{\Gamma}_n\right) = \left(\frac{k}{2 \pi}\right)^{d-1} \int dX \, W_{\Gamma_0} (X) W_{\Gamma_n}(X)\, .$$ We now make the approximation used in (\[FP\]) that, on averaging, the fluctuations in $A(n)$ are removed and that it may be approximated by its classical or ray-dynamical analogue, the classical phase space autocorrelation function $$\label{tcorr}
\langle A(n)\rangle \approx A_{\rm cl}(n) = \int {\rm d}X\;  \rho_n(X)\rho_0(X)\, .$$ Here, $\rho_0$ is the correspondingly averaged classical phase space density of the source and $\rho_n = {\cal L}^n\rho_0$ its $n$th-reflection iterate. Then $$\langle {\rm Tr}( \hat{K}^2)\rangle \approx \frac{1}{1- v^4}
\sum_{n=-\infty}^\infty v^{2|n|} {A}_{\rm cl} (n).$$ For strongly chaotic systems, the autocorrelation function decays exponentially as $$\label{corr}
A_{\rm cl}(n) \sim a_0 + a_1\exp(-\gamma_{\rm cor} n),$$ where $\gamma_{\rm cor}$ denotes the correlation exponent and $a_0, a_1$ are constants depending on the initial density $\rho_0$ ($\e^{-\gamma_{\rm cor}}$ typically being the 2nd largest eigenvalue of the FP operator.) Note that, for strong damping, it suffices to know the autocorrelation function $A(n)$ for relatively small $n$ in order to 