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
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.