In Chapter 2,
the noise power spectrum
of the generalized `force' on the parameter
was identified as proportional to the dissipation rate due to driving that
parameter at frequency
.
Restricting to that case of a single particle Hamiltonian
in a deformable potential field,
it would be interesting to know if there can exist
`special' deformations (those
which give vanishing
as the frequency
),
other than the class of dilations, translations and rotations identified
in Chapter 3.
I will restrict myself to systems with `hard chaos'
(see Section 3.4),
where
all dynamical correlation functions die exponentially
at long times
,
in other words they are short range.
The moments of (I omit subscripts
) are defined by
![]() |
(D.2) |
For a generic deformation, the
moments ,
,
will
numbers with no particular reason to take the value zero.
The smallest
with
will determine the
dominant power-law
seen as
; in this
case it will be
.
Now a `special' deformation has the property that
the (even) moments
vanish, giving the dominant
power-law
with
an even integer greater than zero.
In this way, we see that the special nature of certain deformations
in a hard-chaos billiard is not due to any long-time conspiracy
in
(all correlations are lost beyond
),
rather to short-time correlations with vanishing
lower moments.
The basic issue is whether
can be written as
an exact time derivative:
if it can, we have a `special' deformation.
We will formalize this more carefully.