Chapter 4: Improving upon the white noise approximation: a new `wall formula'

For calculating the rate of energy absorption due to time-dependent deformation of the confining potential, in this chapter I introduce an improved version of the wall formula. The formulation takes into account the `special' class of deformations that cause no heating in the zero-frequency limit, which was identified in the previous chapter. Recall that since calculation of the exact kernel (3.5) is very complicated, we are interested in an approximate prediction for , and for the noise intensity in particular. From this follows the friction coefficient , according to the recipe in Section 2.1.5.

The simplest estimate for
is
the white noise approximation (WNA) introduced in the previous chapter,
and it leads
(for a 3D cavity) to the well known `wall formula' [29]

where the subscript implies that we are considering a microcanonical ensemble , the number of particles is , and the volume of the cavity is . The deformation of the cavity is described by . A general can be handled simply by replacing by the enesemble average particle speed . The above version of the wall formula was originally derived for the purpose of calculating the so-called one-body dissipation rate in nuclei. The original derivation of this formula is based on a simplified kinetic (gas particle) picture [29]. For an alternate derivation using linear response see [118,120]. Cohen[46] provides the generalization to any dimension .

The main purpose is to introduce an improved version of the wall formula, in the form of an improved estimate for , which I will call the `IFIF' (Section 4.3). This improvement involves projecting out the special components of a general deformation, and only then to estimate using the WNA. This will give an estimate which handles many forms of better than the plain WNA, as I demonstrate numerically.