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
[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
, 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.