Now I explain the observed 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 , where is a constant vector that defines a direction in space. We can write where . A similar relation holds for dilation with . It follows that , where is the power spectrum of . If is a bounded function (as it must be when correlations are short-range), it immediately follows that . Moreover since is a simple function of the particle position, we can assume it is a fluctuating quantity that looks like white noise on timescales . It follows that is generically characterized by behavior for either translations or dilations.
I now consider the case of rotations. For rotations we have , and we can write , where , is a projection of the particle's angular momentum vector 3.2. Consequently . Assuming the angular momentum is a fluctuating quantity that looks like white noise on timescales , we expect that and that is generically characterized by behavior.
Thus we have predictions for the power-laws in the regime for special deformations (assuming hard chaos). This contrasts the generic case of tending to a constant, that is, behavior. These power laws are demonstrated in Fig. 3.6, and have been numerically verified over more than 4 decades in . For an estimate of the prefactor for the dilation case, see Section 6.1.2.
For special deformations we have in the limit , and consequently the dissipation coefficient vanishes (). 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 while keeping constant corresponds to constant deformation velocity (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 .