Next: Energy spreading periodic driving Up: Review of classical dissipation Previous: Adiabatic invariance of phase-space

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

 (2.6)

The external work done in changing the parameter from 0 to can be found by integrating force over distance to give , which must be converted to an increase in the particle energy. Writing this as a time integral, the particle's energy difference at time from the adiabatic value is
 (2.7)

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
 (2.8) (2.9)

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
 (2.10) (2.11)

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
 (2.12)

(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

 (2.13)

where is the (zero-frequency) noise intensity'' of the fluctuating force, equal to the first moment of the autocorrelation function:
 (2.14)

The correlation power spectrum or spectral density of is called , and is the fourier transform of the autocorrelation function. I use the convention
 (2.15)

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 Markovian [79]. Note that, in this picture, if ever becomes comparable to then the diffusion rate should be treated as parameter-dependent .

Next: Energy spreading periodic driving Up: Review of classical dissipation Previous: Adiabatic invariance of phase-space
Alex Barnett 2001-10-03