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 -dimensional space, and is entirely defined by the location of its closed, hard wall-- `driving' will means the parameter moves, or deforms, this wall, according to a `deformation function' .
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 involved, as well as on the amplitude and the driving frequency . Also the number of particles and their energy distribution 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 is proportional to a correlation power spectrum , in both classical and quantum linear response. Hence , also known as the `band profile', will now be the main object of study. Of particular interest is its zero-frequency limit . 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 . 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 , that is the speed of wall movement should be much less than the particle speed at the energy .
I will introduce the white noise approximation (WNA), which uses a strong chaos assumption to give an estimate for . In the nuclear application () this leads to the so-called `wall formula'. I will then compare computed 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 vanishing as various powers of 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:
The Appendices B and C detail the numerical methods used for classical and quantum band profile calculations in this and the following two chapters.