Decomposition of general deformations

The failure of the WNA for `special' deformations also extends to the much wider class of deformations which are similar to special. This is demonstrated in Fig. 4.1. It should be emphasized that this failure happens even if the cavity is strongly chaotic.

We seek an analytical estimate for $\tilde{C}(\omega)$, and in particular for its zero-frequency limit $\nu$. This estimate should apply to any (general) deformation, including the case of `close-to-special' deformations. It would be useful to regard any general deformation as a combination of `special' component and `normal' component. The formulation of this idea is the theme of the present section. Supporting numerical evidence is gathered in the next section.

The special deformations (for which we have $\nu=0$) constitute a linear space, meaning that any sum of special deformations is also a special one. Now we would like to conjecture that there is also a linear space of `normal' deformations. By definition, for `normal' deformation ${\mathcal{F}}(t)$ looks like an uncorrelated random sequence of impulses, and consequently the WNA is a reasonable approximation. The notion of randomness can be better formulated as in Appendix F leading to Eq.(F.4). However in practice (F.4) is not useful, because it cannot be applied as an actual classification tool. (Eq.(F.4) is never satisfied exactly). Still we are going to demonstrate that there is a unique way to identify the subspace of normal deformations, if we insist on a maximal (i.e. the most inclusive) definition of this subspace.

It is important to clarify the heuristic reasoning for having a linear space of normal deformations. The ${\mathcal{F}}(t)$ that corresponds to some normal deformation $D({\mathbf s})$ looks like white noise. It means that only self-correlations of its spikes are statistically significant. If we have two such generic quantities, say ${\mathcal{F}}_1(t)$ and ${\mathcal{F}}_2(t)$, then we expect ${\mathcal{F}}_1(t) + {\mathcal{F}}_2(t)$ to share the same property.

The correlation function of ${\mathcal{F}}(t) = {\mathcal{F}}_1(t) + {\mathcal{F}}_2(t)$ can be written formally as

$$C_{1+2}(\tau) = C_1(\tau) + C_2(\tau) + 2C_{1,2}(\tau)$$ (4.2)

where $C_{1,2}(\tau)$ is the cross-correlation function. In Appendix E we argue the following

$$\int_{-\infty}^{\infty} {C}_{1,2}(\tau) d\tau \ \ = \ \ 0 \hspace*{3cm}$$

$$\mbox{if 1$=$general, 2$=$special}$$ (4.3)

This can also be proved easily using the fact that $\nu_{{\mbox{\tiny E}}}$ is an exact quadratic form (3.5) in the function space of $D{({\mathbf s})}$. (Consider that the special deformations are eigenvectors of this quadratic form with zero eigenvalue, i.e. they lie in the null-space [188]). The result is exact, and does not involve any approximation. In Appendix F we argue the following

$$C_{1,2}(\tau) \; \approx \; c \times \left[ \oint \! D_1({\mathbf s})D_2({\mathbf s}) d{\mathbf s} \right] \delta(\tau) \hspace*{0.5cm}$$

$$\ \ \ \ \ \mbox{if 1$=$normal, 2$=$general}$$ (4.4)

where $c=2m^2 v_{{\mbox{\tiny E}}}^3 \langle\vert\cos\theta\vert^3\rangle/{\mathsf V}$. This result is an approximation, which is expected to be as good as our assumption regarding the `normality' of the deformation $D_1({\mathbf s})$. Consider now the case where $D_1({\mathbf s})$ is normal and $D_2({\mathbf s})$ is special. Both Eq.(4.3) and Eq.(4.4) should apply. But these equations are consistent if and only if $D_1({\mathbf s})$ is orthogonal to $D_2({\mathbf s})$. We say that $D_1({\mathbf s})$ and $D_2({\mathbf s})$ are orthogonal ($1\perp2$) using the following definition:

$$\mbox{orthogonality} \ \ \ \ \ \Leftrightarrow \ \ \ \ \ \oint D_1({\mathbf s})D_2({\mathbf s}) d{\mathbf s} = 0$$ (4.5)

Thus we have proved that normal deformations must be orthogonal (in the sense of (4.5)) to special deformations. Obviously we have proved here a necessary rather than a sufficient condition for `normality'. However, if we insist on a maximal definition for the subspace of normal deformations, then we get a unique identification. Namely, a deformation is classified as `normal' if it is orthogonal to the subspace of special deformations.

The practical consequences of Eq.(4.3) and Eq.(4.4) are as follows:

$$\nu_{1+2} = \nu_{1} \ \ \ \ \ \mbox{if 1$=$general, 2$=$special}$$ (4.6)

and

$$\nu_{1+2} \approx \nu_{1} + \nu_{2} + 2c\oint \! D_1({\mathbf s})D_2({\mathbf s}) d{\mathbf s}$$

$$\ \ \ \ \ \mbox{if 1$=$normal, 2$=$general}$$ (4.7)

These results are tested in the next section.

Figure 4.2: Addition of two `good' normal deformations (1=P2, 2=WG). The two are orthogonal in the sense of (4.5). That they are `good' can be seen by their good agreement with their WNA results (horizontal arrows). The power spectrum of the sum agrees well with the sum of the power spectra.

Figure 4.3: Addition of two `bad' normal deformations (1=FR, 2=SX). The two are orthogonal in the sense of (4.5). That they are `bad' is shown by a lack of agreement with their WNAs. The power spectrum of the sum is badly approximated by the sum of the power spectra (non-linear addition).

Figure 4.4: Addition of a `good' normal deformation (1=WG) to a general deformation (2=SX). The two are orthogonal in the sense of (4.5). The power spectrum of the sum agrees well with the sum of the power spectra.

Figure 4.5: Addition of a general deformation (1=FR) to a `special' deformation (2=TX). The power spectrum of the sum coincides with the sum of the power spectra in the limit $\omega\rightarrow0$, as implied by Eq.(4.6).