Band profile power laws

Now I explain the observed $\nu_{{\mbox{\tiny E}}}=0$ 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 ${\mathbf D}={\mathbf e}$, where ${\mathbf e}$ is a constant vector that defines a direction in space. We can write ${\mathcal{F}}(t)=(d/dt)^2 {\mathcal{G}}(t)$ where ${\mathcal{G}}(t)=-m{\mathbf e}\cdot{\mathbf r}$. A similar relation holds for dilation ${\mathbf D}={\mathbf r}$ with ${\mathcal{G}}(t)=-\mbox{\small$\frac{1}{2}$}m {\mathbf r}^2$. It follows that $\tilde{C}(\omega)=\omega^4\tilde{C}_G(\omega)$, where $\tilde{C}_G(\omega)$ is the power spectrum of ${\mathcal{G}}(t)$. If $\tilde{C}_G(\omega)$ is a bounded function (as it must be when correlations are short-range), it immediately follows that $\tilde{C}(0) = 0$. Moreover since ${\mathcal{G}}(t)$ is a simple function of the particle position, we can assume it is a fluctuating quantity that looks like white noise on timescales $> t_{{\mbox{\tiny erg}}}$. It follows that $\tilde{C}(\omega)$ is generically characterized by $\omega^4$ behavior for either translations or dilations.

I now consider the case of rotations. For rotations we have ${\mathbf D}={\mathbf e}\times{\mathbf r}$, and we can write ${\mathcal{F}}(t)=(d/dt) {\mathcal{G}}(t)$, where ${\mathcal{G}}(t)=-{\mathbf e}\cdot({\mathbf r}\times{\mathbf p})$, is a projection of the particle's angular momentum vector ^3.2. Consequently $\tilde{C}(\omega) = \omega^2\tilde{C}_G(\omega)$. Assuming the angular momentum is a fluctuating quantity that looks like white noise on timescales $> t_{{\mbox{\tiny erg}}}$, we expect that $\tilde{C}(0) = 0$ and that $\tilde{C}(\omega)$ is generically characterized by $\omega^2$ behavior.

Thus we have predictions for the power-laws in the regime $\omega < 1/t_{{\mbox{\tiny erg}}}$ for special deformations (assuming hard chaos). This contrasts the generic case of tending to a constant, that is, $\omega^0$ behavior. These power laws are demonstrated in Fig. 3.6, and have been numerically verified over more than 4 decades in $\omega$. For an estimate of the prefactor for the dilation case, see Section 6.1.2.

For special deformations we have $\tilde{C}(\omega)=0$ in the limit $\omega=0$, and consequently the dissipation coefficient vanishes ($\mu=0$). 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 $\Omega\rightarrow0$ while keeping $A\Omega$ constant corresponds to constant deformation velocity ($\dot{x} =$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 $\mu=0$.

Figure 3.7: The quantum band profile in the 2D quarter stadium at $k \approx 400$ for three choices of deformation: dilation (DI), a generic deformation (G), and a generic deformation restricted to parallel displacement of the stadium upper edge (Gp). In each case, the solid line is the quantum band profile (estimation error 10%); the classical is also shown with a dashed line for comparison. I normalized G and Gp so that they share the same $\nu_{{\mbox{\tiny E}}}^{{\mbox{\tiny WNA}}}$ estimate as DI; this is shown as a horizontal dotted line. The system-specific peak due to `bouncing-ball' orbits is labelled (bb). The inset is a log-log plot with quantum band profile shown as points.