Now I explain the observed
for special deformations,
using classical considerations.
The interested reader should also consult Appendix D.
I start with the case of translations and dilations.
For translations we have
,
where
is a constant vector
that defines a direction in space. We can write
where
.
A similar relation holds for dilation
with
. It follows that
,
where
is the power spectrum
of
.
If
is a bounded function (as it must
be when correlations are short-range),
it immediately follows that
.
Moreover since
is a simple
function of the particle position, we can assume it is a
fluctuating quantity that looks like white
noise on timescales
.
It follows that
is
generically characterized by
behavior
for either translations or dilations.
I now consider the case of rotations.
For rotations we have
,
and we can write
,
where
,
is a projection of the particle's angular momentum vector
3.2.
Consequently
.
Assuming the angular momentum is a
fluctuating quantity that looks like white
noise on timescales
,
we expect that
and that
is
generically characterized by
behavior.
Thus we have predictions for the power-laws in
the regime
for special deformations
(assuming hard chaos).
This contrasts the generic case of
tending to a constant, that is,
behavior.
These power laws are demonstrated in Fig. 3.6,
and have been numerically verified over more than 4 decades in
.
For an estimate of the prefactor for the dilation case, see
Section 6.1.2.
For special deformations we have
in the limit
,
and consequently the dissipation
coefficient vanishes (
).
It should be noted that for the case of
a general combination of translations and rotations
this result
follows from a simpler argument
(one which does not rely on the LRT assumption considered in
[120,118]).
Taking
while keeping
constant corresponds to
constant deformation velocity (
const).
Transforming the time-dependent Hamiltonian into the reference
frame of the cavity (which is uniformly translating
and rotating with constant velocity)
gives a time-independent Hamiltonian.
In the new reference frame the energy is a constant
of the motion, which implies that the system
cannot absorb energy (no dissipation effect),
and hence we must indeed have
.
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