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##

Regime of applicability of linear response

In the adiabatic limit
an initial state
will remain in the local eigenstate
for all .
Hence for extremely small , LRT will not be valid because
Landau-Zener transitions (avoided crossings)
become the dominant heating mechanism, as
explained by Wilkinson [200].
This is called the *quantum adiabatic* regime.
For a given system and given parameter , there will be a typical
change in which results in each energy level encountering
about one avoided crossing.
More strictly this is expressed by the criterion
that eigenstate overlaps (see Section 6.4)
should mix about one nearest neighbour.
This parameter change is called
.
If this change takes longer than
to occur, then the
system remains `localized' on the initial state, and we have adiabaticity.
The above also requires that no dephasing processes occur before
.

LRT also has an upper velocity limit, beyond which non-perturbative
effects dominate.
The key criterion is whether LRT is valid for a single
correlation time
(the timescale required to establish diffusive
spreading [46]).
Once this is true, stochastic energy spreading is expected to continue forever.
FOPT will break down
when there is non-perturbative mixing between levels.
If the band profile is flat near , this happens first
between neighbouring levels, when reaches
.
Therefore the limit of applicability of FOPT is
.
This was once thought [202] to be the point at which
the heating rate departs from LRT.
Beyond this, a `core' region will be created
which is non-perturbatively mixed, before diffusive growth is established.
However, Cohen[46] has realised that the existence of
this core need not invalidate
the LRT result, because it is the `tail' region (everything outside the core)
which dominates the energy spreading.
His modification extends the applicability of LRT up to
, where
is the timescale for much of the probability to have left the initial
level (beyond which perturbation theory is past rescue).

Beyond this, the LRT picture breaks down. However, at the highest ,
it is expected that there is semiclassical
correspondence, so that the *classical* dissipation rate
applies.
The above sequence of spreading profiles,
FOPT core-tail semiclassical, is demonstrated
in our work [48] (the calculations for which appear in
Section 6.4 of this thesis).

** Next:** Quantum-classical correspondence
** Up:** Review of the linear
** Previous:** Relation to susceptibility
Alex Barnett
2001-10-03