Returning now to the FGR expression (2.45), one might wonder how it is possible to get dissipation (irreversible growth in the mean energy) at all, given that the up and down transition rates are always equal! The answer will be that heating is possible because the density of states may differ slightly at the two energies being `pumped' into, and therefore more states may fall under the upper delta function than the lower, giving an increase in mean energy. However to understand this in more detail, the up and down transitions (2.45) must be interpreted as giving diffusion in energy-space. A diffusion equation for time-evolution of a continuous energy distribution function can then be written--from this follows the rate of increase of mean energy.
The linear growth of
resulting from the FGR
means a linear growth in
the second moment (variance) of the distribution about the initial
So far the evolution of the coefficients has been entirely coherent (this is unaffected by the final averaging over initial state ). Therefore an important issue arises: does the energy diffusion eventually stop at , when quantum recurrences due to the discreteness of the spectrum will surely occur? (For instance, in the kicked rotator system , periodic driving results in saturation of energy spreading, giving localization). The answer (discussed in detail in Section 8 and Appendix B of ) is that preservation of coherent effects is actually very fragile. Either can have some slight stochasticity (frequency jitter), or there is some slight dephasing process (interaction with the environment, universal in real systems). Correlations due to coherence are then lost on timescales longer than that associated with jitter or dephasing. So if diffusive spreading and loss of coherence can be established before , the `breaktime' (i.e. time limit of applicability) of the FGR, then stochastic spreading will continue for ever. In other words the `rate equation' (2.49) will remain valid, and we will not be surprised by recurrence (interference) effects at long times. The FGR breaktime will be discussed more in Section 2.2.6.