Beyond the WNA

It is possible now to consider the case of general deformation, and to go beyond the WNA. Given a general deformation $D({\mathbf s})$ we should project out (subtract) all the special components, leaving the normal component, and only then apply the WNA. In Fig. 4.8 we demonstrate this decomposition for the deformation (CO + W16) and the deformation SX. Here and elsewhere, all boundary deformation function integrals (of the form (4.5)) are evaluated using the technique of Appendix G.

The special deformations constitute a linear space which is spanned by the basis functions: one dilation, $d$ translations, and $d(d-1)/2$ rotations. (For $d{=}2$ they are listed in Table 3.2). For a general cavity shape these basis functions are not orthogonal. However, because they are linearly independent, we can use standard linear algebra to build an orthonormal basis $\{ D_i({\mathbf s}) \}$ of special deformations. The special ($\parallel$) and the normal ($\perp$) components of any given deformation $D({\mathbf s})$ are therefore

$$D_{\parallel}({\mathbf s}) = \mbox{$\sum_i \alpha_i D_i({\mathbf s})$}$$

$$D_{\perp}({\mathbf s}) = D({\mathbf s})-D_{\parallel}({\mathbf s})$$ (4.9)

where the coefficients are

$$\alpha_i \; = \; \oint \! D({\mathbf s}) D_i({\mathbf s}) \, d{\mathbf s}.$$ (4.10)

The improved approximation for $\nu$ applies the WNA only to the normal component, giving

$$\nu_{{\mbox{\tiny E}}} \ \approx \ 2m^2 v_{{\mbox{\tiny E}}}^3\l... ...rt^3\rangle \frac{1}{{\mathsf V}} \oint d{\mathbf s} [D_{\perp}({\mathbf s})]^2$$ (4.11)

which we name the IFIF (Improved Fluctuations Intensity Formula). In the particular case of $d{=}3$, substitution of this result into the microcanonical FD relation gives an `improved wall formula' consisting of the replacement of $D{({\mathbf s})}$ by $D_{\perp}{({\mathbf s})}$ in Eq. (4.1).