The white noise approximation (WNA)

The most naive estimate of the fluctuations intensity is based on the `white noise approximation' (WNA). Namely, one assumes that the correlation between bounces can be neglected. This corresponds [120] to the local part of the kernel (3.5). In such a case only the self-correlation of the spikes $C_{{\mbox{\tiny E}}}^{{\mbox{\tiny self}}}(\tau)$ is taken into consideration. Using (3.2) and (3.4) one obtains

$$\nu_{{\mbox{\tiny E}}}\ \approx \ \nu_{{\mbox{\tiny E}}}^{{... ...) \ D_i^2 \ \delta(t-t_i) \right\rangle_{{\mbox{\tiny E}}} ,$$ (3.11)

where the microcanonical average can then be replaced by the time average. (The same expression also results from considering the time-averaged contribution of the shaded parts in Fig. 3.3b). From here the result (3.10) can be used, giving the WNA result

$$\nu_{{\mbox{\tiny E}}}^{{\mbox{\tiny WNA}}} \ = \ 2m^2 v_{{\mbox... ...eta\vert^3\rangle \frac{1}{{\mathsf V}} \oint [D({\mathbf s})]^2 d{\mathbf s} ,$$ (3.12)

where the geometric factor $\langle\vert\cos\theta\vert^3\rangle$ is given above. If we can use the convention $\vert D({\mathbf s})\vert \sim 1$ over the deformed region (and zero otherwise), then we can write the WNA as $\nu_{{\mbox{\tiny E}}}^{{\mbox{\tiny WNA}}} = (2mv_{{\mbox{\tiny E}}})^2 \times (1/\tau_{{\mbox{\tiny col}}})$ which defines $(1/\tau_{{\mbox{\tiny col}}})$ as the effective collision rate (for more discussion of this convention see Appendix F of [46], and [48]). Again, $\tau_{{\mbox{\tiny col}}}$ can be much larger than the ballistic time $\tau_{{\mbox{\tiny bl}}}$ in the case that only a small piece of the boundary is being deformed.

For hard walls, the band profile $\tilde{C}_{{\mbox{\tiny E}}}(\omega)$ is non-zero for all omega, corresponding to an infinite bandwidth of the matrix $(\partial {\mathcal{H}} / \partial x)_{nm}$. The band profile at $\omega \gg \tau_{{\mbox{\tiny bl}}}^{-1}$ is entirely due to the `self' component, so is constant and given by the WNA (see Fig. 3.3d). This convergence at large $\omega$ has been tested numerically: the decay to the constant WNA value is found to be quite oscillatory.

The use of the WNA can be justified whenever successive collisions are effectively uncorrelated. The applicability of such an assumption depends on the shape of the cavity (which will determine the decay of correlations via the typical Lyapunov exponent) as well as on the type of deformation involved. If we have the cavity of Fig. 3.2a, and the deformation involves only a small piece of the boundary (e.g. see Fig. 3.2b), then successive collisions with the deformed part of the boundary are effectively uncorrelated. This is so because there are many collisions with static pieces of the boundary before the next effective collision (with non zero $D_i$) takes place. If the deformation involves a large piece (or all) of the boundary, one can still argue that successive collisions are effectively uncorrelated provided $D{({\mathbf s})}$ is `oscillatory' enough (i.e. changes sign many times along the boundary).

These expectations are qualitatively confirmed by the numerical results of Fig. 3.4, where I show a sequence of deformations for which the WNA performs increasingly well. The numerical method and estimation error for finding the classical $\tilde{C}_{{\mbox{\tiny E}}}(\omega)$ are presented in Appendix B.