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 $\tilde{C}_{{\mbox{\tiny E}}}(\omega)$, and for the noise intensity $\nu_{{\mbox{\tiny E}}}$ in particular. From this follows the friction coefficient $\mu$, according to the recipe in Section 2.1.5.

The simplest estimate for $\nu_{{\mbox{\tiny E}}}$ 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]

$$\mu_{{\mbox{\tiny E}}} = \frac{N}{\mathsf{V}} m v_{{\mbox{\tiny E}}} \oint D({\mathbf s})^2 d{\mathbf s}$$ (4.1)

where the subscript ${\mbox{\tiny E}}$ implies that we are considering a microcanonical ensemble $\rho(E)$, the number of particles is $N$, and the volume of the cavity is ${\mathsf{V}}$. The deformation of the cavity is described by $D{({\mathbf s})}$. A general $\rho(E)$ can be handled simply by replacing $v_{{\mbox{\tiny E}}}$ by the enesemble average particle speed $\bar{v}$. 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 $d$.

The main purpose is to introduce an improved version of the wall formula, in the form of an improved estimate for $\nu_{{\mbox{\tiny E}}}$, 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 $\nu_{{\mbox{\tiny E}}}$ using the WNA. This will give an estimate which handles many forms of $D{({\mathbf s})}$ better than the plain WNA, as I demonstrate numerically.

Figure 4.1: The failure of the WNA estimate for $\tilde{C}(\omega)$ for deformation types CO (similar to DI) and SX (similar to TX). The WNA is clearly a vast overestimate of the small-$\omega$ limit. See Tables 3.1 and 3.2 for explanation of deformation types.