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Energy spreadingperiodic driving case
We can build on the results of the previous section to
see how things change if the parameteric driving
is sinusoidal:
.
We will find the new spreading rate
in the same fashion.
For a general , Eq. (2.7) becomes

(2.16) 
As before, squaring this and taking the microcanonical average gives
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 range expanded as before to
,
appropriate when
.
The final line shows an oscillating term
and a diffusive term (linear in ),
and is valid for any .
The diffusion
becomes dominant in the limit
corresponding
to observation over a large number of parameter oscillation periods
(the fractional error due to the oscillating term dying like
),
giving linear growth of the second moment.
This longtime diffusion rate is

(2.18) 
where is now defined as the mean square parameter velocity
.
So it is apparent that the diffusion is determined by the
spectral density of
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
while
is held constant, using
,
and noticing that
the two terms in (2.17) now contribute equally.
Next: FokkerPlanck equation for energy
Up: Review of classical dissipation
Previous: Energy spreading constant velocity
Alex Barnett
20011003