Energy spreading--constant velocity case

We now derive the next-order correction to the adiabatic limit,
namely spreading in energy (or equivalently in ),
which causes the
*irreversible* heating.
This was first found by Ott [153] using `multiple-time-scale analysis'.
To derive the spreading rate,
we now assume
constant parameter velocity over a time interval
long enough to establish diffusive behaviour
(following [47,46]).
The longer-time evolution for a general can
then be built from these short constant-velocity segments.

For any given particle trajectory
(ensemble member
launched at
at )
there is an associated stochastic time-dependent `force'
on the parameter .
The time-dependence arises from that of the trajectory.
In the billiard case, this is simply the impulses
the particle exerts on the deforming part of the wall.
We extract the fluctuation of this quantity about
its average (which from (2.3) is just the conservative
force ), giving the definition

Now has a finite correlation time which is similar to or less than the ergodic time. For then the above expression is simply the final energy change resulting from a `random walk' of step size . Squaring this and taking a microcanonical average (at energy ) gives the energy variance

This last approximation is good for , where the double integral grows linearly in (with fractional error from linearity dying like ). This is simply the variance of a random walk growing linearly in time. It can also be shown using a transform of variables to and , with the range of being , and the range of can be taken to , as illustrated in Fig. 2.2a. We will write in terms of , the autocorrelation function of the fluctuating force. depends on both the classical motion at energy and on the particular deformation chosen, and is defined by

where the second equality states the assumption that the autocorrelation does not change over the timescale . This latter condition restricts to be much smaller than , the parametric change which changes the classical Hamiltonian (and hence the statistical properties of the system) significantly. In combination with the limit , this gives

(this is the `trivial slowness condition' of Cohen [46]).

We now have established diffusive energy spreading
with a rate
(given by substitution of (2.10) into (2.8))
of

The

Now that diffusion is established for short time-steps , the evolution over longer times with a general can be composed of independent diffusive steps each operating on the probability distribution given by the previous step. This `memory-less' stochastic approximation is called