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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

$\displaystyle D_{\parallel}({\mathbf s})$ $\textstyle =$ $\displaystyle \mbox{$\sum_i \alpha_i D_i({\mathbf s})$}$  
$\displaystyle D_{\perp}({\mathbf s})$ $\textstyle =$ $\displaystyle D({\mathbf s})-D_{\parallel}({\mathbf s})$ (4.9)

where the coefficients are
\begin{displaymath}
\alpha_i \; = \; \oint \! D({\mathbf s}) D_i({\mathbf s}) \, d{\mathbf s}.
\end{displaymath} (4.10)

The improved approximation for $\nu$ applies the WNA only to the normal component, giving
$\displaystyle \nu_{{\mbox{\tiny E}}} \ \approx \
2m^2 v_{{\mbox{\tiny E}}}^3\l...
...rt^3\rangle
\frac{1}{{\mathsf V}}
\oint d{\mathbf s} [D_{\perp}({\mathbf s})]^2$     (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).



Subsections
next up previous
Next: Numercial tests of the Up: Chapter 4: Improving upon Previous: Symmetry effects
Alex Barnett 2001-10-03