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FokkerPlanck equation for energy evolution
Stochastic evolution is generally governed by a FokkerPlanck equation,
describing
drift and diffusion of a density function (see Appendix A
and [79]).
The derivation of the correct FokkerPlanck equation
for the timeevolution of the
energy distribution
is not trivial.
Now that we have the energy diffusion rate
at a given energy , it is
tempting to use it directly as a standard (energydependent)
diffusion term
.
This simple answer is wrong, and would give the wrong heating rate.
As discussed in Appendix A, care is needed in
writing the diffusion term.
The equation was first written down correctly (without derivation)
by Wilkinson [201].
Jarzynski [106] derives
it in a clear fashion (which I will follow), and later he [108]
and Berry and Robbins
[27] rederive it using expansions
of the full phasespace density in powers of a `slowness parameter'
().
It must be realised that, although the spreading rate was derived in energy,
it is that is the `natural' variable in which to discuss spreading.
Not only is the evolution of
simple,
but the consideration of Liouville's theorem becomes obvious.
In the adiabatic limit we have seen that
is constant (2.5).
At finite the evolution generally becomes

(2.19) 
Liouville's equation of motion [126] for
,
the full phasespace probability density, is
.
Regardless of the Hamiltonian and its timedependence,
this means that a constant
density must remain so for all times.
Recalling that is just the average of over an energyshell,
this means
must
be a solution of (2.19).
It must also be a solution
with zero flux everywhere in
(if this we not so, a discontinuity
would arise at ; see also Appendix B of [106] ).
This implies that in (2.19) the drift term is zero,
and the diffusion is unbiased in space.
By `unbiased' I mean that at every value of ,
zero density gradient results in zero flux.
Note that the may still depend on .
Now this ``Liouville trick'' has been performed, we can transform
the evolution back to energyspace;
this is done using Appendix A, and gives
a FokkerPlanck equation with nonzero drift,

(2.20) 
The first term gives the usual adiabatic energy change, while the second
is a drift term entirely due to diffusion.
This gives the final energy evolution, written with the
terms
grouped together to give a modified diffusion term,

(2.21) 
Remember that (2.21)
is an effective equation for timescales longer than
the ergodic time.
In the context of billiards, energy evolution
can be thought of as the
biased `random walk' in energy space due to the
sum of `kicks' due to the particle hitting the moving walls [107].
Next: Irreversible growth of average
Up: Review of classical dissipation
Previous: Energy spreading periodic driving
Alex Barnett
20011003