Appendix D: How Many Special Deformations Are There?

In Chapter 2, the noise power spectrum $\tilde{C}_{{\mbox{\tiny E}}}(\omega)$ of the generalized `force' on the parameter was identified as proportional to the dissipation rate due to driving that parameter at frequency $\omega$. 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 $\tilde{C}_{{\mbox{\tiny E}}}(\omega)$ as the frequency $\omega\rightarrow0$), 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 $\gg t_{{\mbox{\tiny erg}}}$, in other words they are short range.

The moments of $C(\tau)$ (I omit subscripts ${\mbox{\tiny E}}$) are defined by

$$\mathsf{M}_p \; \equiv \; \int_{-\infty}^\infty C(\tau) \tau^p d\tau, \hspace{1in} \mbox{integer $p\ge 0$},$$ (D.1)

and give the Taylor expansion coefficients of $\tilde{C}(\omega)$ about $\omega=0$, as follows:

$$\left. \left(-i \frac{\partial }{\partial \omega}\right)^p \right\vert _0 \tilde{C}(\omega)\; = \; \mathsf{M}_p$$ (D.2)

which follows from the definition of the fourier transform. Since $C(\tau) = C(-\tau)$, all the odd moments are zero. We will restrict all the even moments to be finite; this is equivalent to saying $\tilde{C}(\omega)$ has a well-defined $\omega=0$ limit, namely $\mathsf{M}_0$, and a well-defined local expansion in non-negative even powers of $\omega$. This will exclude certain $C(\tau)$ with power-law (long range) tails.

For a generic deformation, the moments $\mathsf{M}_0$, $\mathsf{M}_2$, $\mathsf{M}_4 \cdots$ will numbers with no particular reason to take the value zero. The smallest $p$ with $\mathsf{M}_p \ne 0$ will determine the dominant power-law $\omega^\gamma$ seen as $\omega\rightarrow0$; in this case it will be $\gamma = 0$. Now a `special' deformation has the property that the (even) moments $p < \gamma$ vanish, giving the dominant power-law $\omega^\gamma$ with $\gamma$ 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 ${\mathcal{F}}(t)$ (all correlations are lost beyond $t_{{\mbox{\tiny erg}}}$), rather to short-time correlations with vanishing lower moments.

The basic issue is whether ${\mathcal{F}}(t)$ can be written as an exact time derivative: if it can, we have a `special' deformation. We will formalize this more carefully.