Historical remarks and conclusion

A few concluding remarks regarding the history of the wall formula are in order. It has been known since its inception that the naive wall formula gives unphysical answers in the case of constant-velocity translations and rotations. This was first regarded as a kinetic gas `drift' effect [29]. It should be noted that the recipe presented in [29], namely to subtract this drift component, is equivalent in practice to the recipe (4.9) that we have presented here, provided we ignore dilations. It is also important to realize that the argumentation in [29] for this subtraction appears to be ad hoc, being based on a `least-structured drift pattern' reasoning. A stated condition on this subtraction was that the resulting deformation preserve the location of the `center of mass' (centroid) of the cavity, for reasons particular to the nuclear application [29]. This condition seems to have become standard practice in numerical tests of the wall formula [158,149,30,31,32]. However, as Fig. 4.7a shows, this condition is generally not equivalent to the above subtraction of translation and rotation components ^4.1. This seems to invalidate the theorem presented in Section 7.1 of [29]. Where the flaw in their reasoning lies we are not sure.

The consideration of the special nature of dilations is absent from the literature. Even if we restrict ourselves to volume-preserving deformations (the nuclear dissipation case), then deformations of certain cavities can be found for which the dilation correction is significant. This correction can only be large if the cavity has a large variation in radius (i.e. is highly non-spherical). We illustrate this in Fig. 4.7b. We suggest this as a possible reason why major discrepancies due to dilation have not emerged in the numerical tests of the wall formula until now. Such tests have generally been of shapes close to a 3D sphere [29,158,149,30,31,32].

Hence we believe that the new recipe presented, along with the associated theory and in conjunction with the particular power-law dependences from the previous chapter, is a significant step in the treatment of one-body dissipation and of dissipation in $d$-dimensional cavities in general.