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Relation to susceptibility
Here, for the sake of intuition, I briefly make contact with the traditional
LRT notation arising in condensed matter physics
[127,122,160,99,65].
I consider first the `response' (timedependent expectation value)
of a general measurable whose operator is .
Looking at Eq.(2.40) we can write

(2.54) 
where the first term is the constant
equilibrium value
and the other terms
have been kept only up to first order in the driving .
The response is linear
but is not generally local in time, i.e. it has memory.
However, because of translational invariance in time, the response must be
local in frequency.
So we can relate the Fourier transform of response to that of driving by
, where
is a generalized susceptibility.
Numerous symmetry relations of
follow from
causality[127]; in particular
.
Substitution of (2.40) gives

(2.55) 
where the latter form is averaged over with the appropriate
initial distribution , and is a standard
result of LRT [65].
We now choose
whereupon the subscript
is
no longer needed.
The susceptibility is split into real and imaginary parts,
.
The dissipation (energy absorption) is determined by [127],
since, using the driving (2.41) we have
where at the last stage the oscillatory terms average to zero.
This heating rate expression results entirely from the definition of .
However now our friction coefficient from the previous section
can be seen to be simply
.
We should also note that in the condensed matter physics context
is proportional to the dynamic form factor
if we associate
with the density operator
of a mode labelled
[160].
In the case where is flux enclosed by a mesoscopic sample (quantum dot),
is a current operator, thus
determines the conductivity of the system.
As one would expect, the conductivity is then given by the
timeintegral of the currentcurrent
correlation
[122,99,8].
We have performed such a calculation for a chaotic mesoscopic dot
[15].
Next: Regime of applicability of
Up: Review of the linear
Previous: Heating rate example noninteracting
Alex Barnett
20011003