The classical calculation of Section 2.1 and the quantum LRT calculation of Section 2.2 gave the same answers for heating rate. This is remarkable because the energy evolution (spreading profile) was different: the classical spreading is gradual (for ), whereas the quantum spreading involves jumps in units of (for driving frequency ). It is also remarkable because the quantum result is therefore independent of . However the heating rates agree because the expressions for spreading rate (second moment of spreading profile) were the same. The only difference is the replacement of the classical definition of by the corresponding quantum-mechanical definition . If these two agree, then there is quantum-classical correspondence (QCC) as far as dissipation (friction ) is concerned [47,46]. Cohen calls this `restricted correspondence' because of the different spreading profiles. Here I present numerical evidence supporting this claim of QCC in a real system (a 2D cavity). Hence in the following chapters I will move freely between the quantum and classical pictures. In particular, the term `band profile' will then refer to both quantum and classical auto-correlation spectra.
First I briefly mention QCC outside the LRT regime (whose limits were outlined in the previous section). For extremely small in the quantum-adiabatic regime, there is no QCC expected because the level spacing distribution for small dominates the heating rate . This is a purely quantum effect (determined by certain quantum symmetries [35,146]), and has no reason to agree with any classical quantity. At the other extreme, as the maximum where LRT is valid also vanishes. Therefore if one is to have QCC in the semiclassical limit (a fundamental requirement of quantum mechanics being that it reduces to classical mechanics in this limit), some new mechanism is required. As explained by Cohen[47,46], in this limit QCC is achieved through semiclassical agreement of the spreading profiles (`detailed correspondence').