Appendix D: How Many Special Deformations Are There?

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.