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
is taken into
consideration.
Using (3.2) and (3.4) one obtains

where the geometric factor is given above. If we can use the convention over the deformed region (and zero otherwise), then we can write the WNA as which defines as the effective collision rate (for more discussion of this convention see Appendix F of [46], and [48]). Again, can be much larger than the ballistic time in the case that only a small piece of the boundary is being deformed.

For hard walls, the band profile is non-zero for all omega, corresponding to an infinite bandwidth of the matrix . The band profile at is entirely due to the `self' component, so is constant and given by the WNA (see Fig. 3.3d). This convergence at large 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 )
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
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 are presented in Appendix B.