Chapter 3: Dissipation rate in chaotic billiards, and `special' deformations

In this chapter I shall be calculating the dissipation rate resulting from driving a chaotic billiard (also known as a cavity) system containing a single particle or gas of non-interacting particles. The billiard system is in $d$-dimensional space, and is entirely defined by the location of its closed, hard wall-- `driving' will means the parameter $x$ moves, or deforms, this wall, according to a `deformation function' $D{({\mathbf s})}$.

What is the rate in which the `gas' inside the cavity is heated up? The answer depends on the shape of the cavity, the deformation $D({\mathbf s})$ involved, as well as on the amplitude $A$ and the driving frequency $\omega$. Also the number of particles $N$ and their energy distribution $\rho(E)$ should be specified. To reach an answer I shall be using the theory outlined in the previous chapter, where it was explained that the dissipation rate due to driving at frequency $\omega$ is proportional to a correlation power spectrum $\tilde{C}_{{\mbox{\tiny E}}}(\omega)$, in both classical and quantum linear response. Hence $\tilde{C}_{{\mbox{\tiny E}}}(\omega)$, also known as the `band profile', will now be the main object of study. Of particular interest is its zero-frequency limit $\nu_{{\mbox{\tiny E}}}$. $\tilde{C}_{{\mbox{\tiny E}}}(\omega)$ will take different forms for the case of different deformations and for different cavity shapes-- I will be interested in general deformations which need not preserve the cavity shape nor its volume. I also assume shapes such that the motion of the particle inside the cavity is globally chaotic, meaning no mixed phase space [154]. The criteria for having such a cavity are discussed in [40,204]. For validity of linear response, the slowness condition of (2.12) is assumed, which in the billiard case becomes $V \ll v_{{\mbox{\tiny E}}}$, that is the speed of wall movement should be much less than the particle speed at the energy $E$.

I will introduce the white noise approximation (WNA), which uses a strong chaos assumption to give an estimate for $\tilde{C}_{{\mbox{\tiny E}}}(\omega)$. In the nuclear application ($d=3$) this leads to the so-called `wall formula'. I will then compare computed $\tilde{C}_{{\mbox{\tiny E}}}(\omega)$ curves to the WNA prediction, and find many deformations for which the WNA fails. In particular, the main result will be the discovery of a class of deformations which have $\tilde{C}_{{\mbox{\tiny E}}}(\omega)$ vanishing as various powers of $\omega$ in the zero-frequency limit, which I name `special' deformations. This class is the set of deformations that are shape-preserving: they involve only translations, rotations and dilations of the cavity. Note that translations and rotations are also volume-preserving, in which case the associated time-dependent deformations can be described as `shaking' the cavity. The special class is important for three reasons:

1. It will provide the basis for the improved WNA estimate (improved `wall formula') for $\nu_{{\mbox{\tiny E}}}$, which takes into account correlations beyond the strong chaos assumption. This is the subject of Chapter 4.
2. The `special' nature of the deformation involving dilation about an arbitrary origin corresponds to quasi-orthogonality of the cavity quantum eigenstates on the boundary. This in turn has a very direct connection to the success of the numerical method of Vergini and Saraceno [195], the subject of Chapter 6.
3. They have other applications: the band profile of certain special deformations is simply related to the band profile for driving the (charged) particle by uniform external fields. For instance, the mesoscopic version of Drude formula for the conductance of a quantum dot in a uniform time-dependent magnetic field reduces to the the calculation of $\tilde{C}_{{\mbox{\tiny E}}}(\omega)$ for the rotation deformation. For more details, I refer the reader to our published work [15].

The Appendices B and C detail the numerical methods used for classical and quantum band profile calculations in this and the following two chapters.