Energy spreading--periodic driving case

We can build on the results of the previous section to see how things change if the parameteric driving is sinusoidal: $x(t)=A\sin(\omega t)$. We will find the new spreading rate $D_{{\mbox{\tiny E}}}$ in the same fashion. For a general $x(t)$, Eq. (2.7) becomes

$${\mathcal{H}}(t) - E(x(t)) = - \int_0^t dt_1 \dot{x}(t_1) {\mathcal{F}}(t_1),$$ (2.16)

As before, squaring this and taking the microcanonical average gives

$$\langle [ {\mathcal{H}}(t) - E(x(t)) ]^2 \rangle_{{\mbox{\tiny E}}} \; =\; (A \omega)^2 \, \int_0^t \int_0^t dt_1 dt_2 \; \cos (\omega t_1) ... ...a t_2) \langle {\mathcal{F}}(t_1) {\mathcal{F}}(t_2) \rangle_{{\mbox{\tiny E}}}$$

$$\approx \frac{(A \omega)^2}{2} \, \int_0^t dt' \int_{-\infty}^\infty d\tau [ \cos(2\omega t') + \cos(\omega \tau) ] \, C_{{\mbox{\tiny E}}}(\tau)$$

$$= \frac{(A \omega)^2}{2} \left( \frac{\sin (2\omega t)}{2\omega} \nu_{{\mbox{\tiny E}}}\: + \: \tilde{C}_{{\mbox{\tiny E}}}(\omega)\cdot t \right) .$$ (2.17)

On the second line above, the same transformation of time variables was used as before (Fig. 2.2a, except now multiplied by the `checkerboard' of the cos functions), and the $\tau$ range expanded as before to $[-\infty,\infty]$, appropriate when $t \gg \tau_{{\mbox{\tiny cl}}}$. The final line shows an oscillating term and a diffusive term (linear in $t$), and is valid for any $\omega$. The diffusion becomes dominant in the limit $\omega t \gg 1$ corresponding to observation over a large number of parameter oscillation periods (the fractional error due to the oscillating term dying like $1/\omega t$), giving linear growth of the second moment. This long-time diffusion rate is

$$D_{{\mbox{\tiny E}}} \; = \; \mbox{\small$\frac{1}{2}$}\tilde{C}_{{\mbox{\tiny E}}}(\omega)V^2$$ (2.18)

where $V^2$ is now defined as the mean square parameter velocity $\langle \dot{x}^2 \rangle = \mbox{\small$\frac{1}{2}$}(\omega A)^2$.

So it is apparent that the diffusion is determined by the spectral density of ${\mathcal{F}}(t)$ at the driving frequency. As this frequency goes to zero, the linear driving result (2.13) is recovered. This can be verified by taking the limit $\omega\rightarrow0$ while $\omega A = \dot{x}$ is held constant, using $\tilde{C}_{{\mbox{\tiny E}}}(\omega\rightarrow0) = \nu_{{\mbox{\tiny E}}}$, and noticing that the two terms in (2.17) now contribute equally.