The WNA revisited--cavity shape effects

In Section 3.2 we have assumed that generic fluctuating quantities such as ${\mathbf r}^2$ and ${\mathbf e}\cdot{\mathbf r}$ and ${\mathbf e}\cdot({\mathbf r}\times{\mathbf p})$ have a white noise power spectrum for $\omega \ll 1/\tau_{{\mbox{\tiny bl}}}$. In section 4.1 we are going to suggest that this white noise assumption is approximately true for any fluctuating quantity ${\mathcal{F}}(t)$ that comes from a normal deformation (the term `normal' will be defined there).

Obviously, the goodness of the `white noise assumption' in the two cases mentioned is related to the chaoticity of the system, and should be tested for particular examples. This has been done so for the cavity of Fig. 3.2 (see [14], and Figs. 3.4 and 4.2). This cavity is an example of a `scattering billiard' and so exhibits strong chaos (no marginally-stable orbits) [40]. If the motion is not strongly chaotic we may get a $C(\tau)$ that decays like a power law (say $1/\tau^{1{-}\gamma}$ with $0<\gamma \le 1$) rather than an exponential [40,78,54,37] (time of crossover to algebraic decay is discussed in [53]). In such case the universal behavior is modified: we get $\omega^{-\gamma}$ behavior for $\tilde{C}_{{\mbox{\tiny E}}}(\omega)$ at small frequencies ( $\nu_{{\mbox{\tiny E}}}$ diverges), signifying faster-than-diffusive energy spreading in Eq.(2.8) [37]. The stadium is an example where such a complication may arise: an ergodic trajectory can remain in the marginally-stable `bouncing ball' orbit family (between the top and bottom edges) for long times, with a probability scaling as $t^{-1}$ [78,54,53]. Depending on the choice of $D{({\mathbf s})}$ this may manifest itself in $C(\tau)$. For example, in Figs. 2.5 and 3.7 the deformations P and Gp respectively both involve a distortion confined to the upper edge, and the resulting sensitivity to the bouncing ball orbit leads to large enhancement of the fluctuations intensity $\tilde{C}(\omega{=}0)$, and is suggestive of singular behavior for small $\omega$. However in the same system the deformations W2 and G, which are zero on the upper and lower edges, show no such enhancement--the band profile is flat as $\omega\rightarrow0$ and the deviation from strong chaos is masked.

If the billiard has a mixed phase space (which is the generic case), then the integrable component does not contribute to diffusive energy spreading. Proposals have been made to account for this via a phase-space volume factor [158,149].