Addition of deformations: numerical tests

On the basis of the discussion in the previous section we define normal deformation as those that are orthogonal to all special deformations, in the sense of Eq. (4.5). Obviously there are `good' normal deformations for which the WNA is an excellent approximation (P1 and W8 in Fig. 3.4, for example), and there are `bad' normal deformations for which the WNA is not a very good approximation (FR in Fig. 3.4, and the normal component in Fig. 4.8b). In this section we present numerical evidence that verifies the theoretical results of the previous section, and investigate how `bad' a normal deformation has to be for them to break down.

From what we have claimed it follows that if $D_1{({\mathbf s})}$ and $D_2{({\mathbf s})}$ are orthogonal normal deformations, then $\nu_{1+2}=\nu_1+\nu_2$. We could as well write

$$\tilde{C}_{1+2}(\omega) \; \approx \; \tilde{C}_1(\omega)+\tilde{C}_2(\omega) \hspace*{2cm}$$

$$\ \ \ \ \ \mbox{if 1{=}normal, 2{=}normal, and 1 $\perp$\ 2}$$ (4.8)

because by assumption the three correlation functions are approximately flat. We demonstrate this addition rule in the case of two `good' deformations which are orthogonal in Fig. 4.2. We found that small `pistons' (P2 is significant on only $\sim 1/50$ of the perimeter) were needed to achieve addition of the accuracy (a few %) shown. However, the restriction on the `wiggle' type of deformation was somewhat more lenient (WG is $\sim 5$ times wider than P2 yet obeys the WNA better than P2 does).

In general we observe that the quality of the addition rule is limited by the deviation from the WNA of the better of the two deformations. In Fig. 4.3 we see that if both $D_1({\mathbf s})$ and $D_2({\mathbf s})$ are bad, then also the addition rule (4.8) becomes quite bad. Fig. 4.4 shows that the addition rule (4.8) is reasonably well satisfied also if either $D_1({\mathbf s})$ or $D_2({\mathbf s})$ is a `good' normal deformation. We have chosen $D_1({\mathbf s})$ as WG (good), and $D_2({\mathbf s})$ as SX which is almost completely dominated by the special x-translation deformation. The addition rule (4.8) is obeyed at all $\omega$. This proves that our assertions Eq.(4.4) about the vanishing of $C^{{\mbox{\tiny non-self}}}_{1,2}(\tau)$ is indeed correct. It holds here as a non-trivial statement ( $D_2({\mathbf s})$ is general and `bad').

Finally, we consider the case where $D_1({\mathbf s})$ is general and $D_2({\mathbf s})$ is special. This is illustrated in Fig. 4.5. The addition rule (4.8) becomes exact in the limit of small frequency corresponding to the vanishing of $\tilde{C}_{1,2}(\omega\rightarrow0)$ as implied by Eq.(4.3). In particular this implies that $\nu_{1+2}=\nu_1$. Note that there is no condition on the orthogonality of $D_1({\mathbf s})$ and $D_2({\mathbf s})$). This will be the key to for improving over the WNA, which we are going to discuss in the next section.